Chapter 5 Thomson Effect, Exact Solution, and Compatibility Factor

Transcription

1 5-1 Chapter 5 homson Effect, Exact Solution, and Compatibility Factor he formulation of the classical basic equations for a thermoelectric cooler from the homson relations to the non-linear differential equation with Onsager s reciprocal relations was performed to basically study the homson effect in conjunction with the ideal equation. he ideal equation is obtained when the homson coefficient is assumed to be zero. he exact solutions derived for a commercial thermoelectric cooler module provided the temperature distributions including the homson effect. he positive homson coefficient led to a slight improvement on the performance of the thermoelectric device while the negative homson coefficient led to a slight declination of the performance. he comparison between the exact solutions and the ideal equation on the cooling power and the coefficient of performance over a wide range of temperature differences showed close agreement. he homson effect is small for typical commercial thermoelectric coolers and the ideal equation effectively predicts the performance. 5.1 homson Effects hermoelectric phenomena are useful and have drawn much attention since the discovery of the phenomena in the early nineteenth century. he barriers to applications were low efficiencies and

2 5- the availability of materials. In 181, homas J. Seebeck discovered that an electromotive force or potential difference could be produced by a circuit made from two dissimilar wires when one junction was heated. his is called the Seebeck effect. In 1834, Jean Peltier discovered the reverse process that the passage of an electric current through a thermocouple produces heating or cooling depended on its direction [1]. his is called the Peltier effect (or Peltier cooling). In 1854, William homson discovered that if a temperature difference exists between any two points of a current-carrying conductor, heat is either absorbed or liberated depending on the direction of current and material []. his is called the homson effect (or homson heat). hese three effects are called the thermoelectric effects. homson also developed important relationships between the above three effects with the reciprocal relations of the kinetic coefficients (or sometimes the so-called symmetry of the kinetic coefficients) under a peculiar assumption that the thermodynamically reversible and irreversible processes in thermoelectricity are separable []. he relationship developed is called the homson (or Kelvin) relations. he necessity for the assumption remained an objection to the theory until the advent of the new thermodynamics [3-5]. he relationship was later completely confirmed by experiments to be essentially a consequence of the Onsager s Reciprocal Principle in 1931 [6]. As supported by Onsager s Principle, the homson relations provide a simple expression for Peltier cooling, which is the product of the Seebeck coefficient, the temperature at the junction, and the current. his Peltier cooling is the principal thermoelectric cooling mechanism. here are two counteracting phenomena, which are the Joule heating and the thermal conduction. he net cooling power is the Peltier cooling minus these two effects. Actually the Joule heating affects the thermal conduction and consequently the Peltier cooling is subtracted only by the thermal conduction. Expressing the net cooling power in terms of the heat flux vector q gives [7]

3 Formulation of Basic Equations Onsager s Reciprocal Relations he second law of thermodynamics with no mass transfer (S = 0) in an isotropic substance provides an expression for the entropy generation. S gen Q 1 (5.1) Suppose that the derivatives of the entropy generation sgen per unit volume with respect to arbitrary quantities xi are quantities Ji [4, 8] as s x gen i J i (5.) At maximum sgen, Ji is zero. Accordingly, at the state close to equilibrium (or maximum), Ji is small [3]. he entropy generation per unit time and unit volume is s gen s t gen xi t J i (5.3) We want to express x i t as X i. hen, we have s gen X i J i (5.4) he X i are usually a function of J i. Onsager [6] stated that if J i were completely independent, we should have the relation, expanding X i in the powers of Ji and taking only linear terms [7]. he smallness of J i in practice allows the linear relations being sufficient.

4 5-4 X i j R ij J j (5.5) where Rij are called the kinetic coefficients. Hence, we have s gen X ij i i (5.6) It is necessary to choose the quantities X i in some manner, and then to determine the Ji. he and Ji are conveniently determined simply by means of the formula for the rate of change of the total entropy generation. X i S X J dv > 0 (5.7) gen For two terms, we have i i i S gen X J X J dv > 0 (5.8) 1 1 And X 1 R11J1 R1J (5.9) X R1J1 RJ (5.10) From these equations, one can assert that the kinetic coefficients are symmetrical with respect to the suffixes 1 and. R1 R 1 (5.11) which is called the reciprocal relations.

5 5-5 Basic Equations Let us consider a non-uniformly heated thermoelectric material. For an isotropic substance, the continuity equation for a constant current gives j 0 (5.1) he electric field E is affected by the current density j and the temperature gradient. he coefficients are known according to the Ohm s law and the Seebeck effect [7]. he field is then expressed as E j (5.13) where is the electrical resistivity and is the Seebeck coefficient. Solving for the current density gives j 1 E (5.14) he heat flux q is also affected by the both field E and temperature gradient. However, the coefficients were not readily attainable at that time. homson in 1854 arrived at the relationship assuming that thermoelectric phenomena and thermal conduction are independent. Later, Onsager supported that relationship by presenting the reciprocal principle which was experimentally proved but failed the phenomenological proof [5]. Here we derive the homson relationship using Onsager s principle. he heat flux with arbitrary coefficients is q C E C 1 (5.15) he general heat diffusion equation is

6 5-6 q q c p t For steady state, we have q q 0 (5.16) (5.17) where q is expressed by [7] q E j j j (5.18) Expressing Equation (5.1) with Equation (5.17) in a manner that the heat flux gradient minus sign, the entropy generation per unit time will be q has a Q 1 S gen 1 q q dv (5.19) Using Equation (5.18), we have S gen E j q dv (5.0) he second term is integrated by parts, using the divergence theorem and noting that the fully transported heat does not produce entropy since the volume term cancels out the surface contribution [7]. hen, we have S E j q dv gen (5.1)

7 5-7 If we take j and q as X1 and X, then the corresponding quantities J 1 and J are the components of the vectors E and Equation (5.15), we have [7]. Accordingly, as shown in Equations (5.9) and (5.10) with 1 E 1 j (5.) E q C1 C (5.3) where the reciprocal relations in Equation (5.11) gives C 1 1 (5.4) hus, we have C1. Inserting this and Equation (5.13) into (5.3) gives q j C (5.5) We introduce the Wiedemann-Franz law that is L o k, where Lo is the Lorentz number (constant). Expressing k as Lo/, we find that the coefficient C is nothing more than the thermal conductivity k. Finally, the heat flow density vector (heat flux) is expressed as q j k (5.6) his is the most essential equation for thermoelectric phenomena. he second term pertains to the thermal conduction and the term of interest is the first term, which gives the thermoelectric

8 5-8 effects: directly the Peltier cooling but indirectly the Seebeck effect and the homson heat [8-10]. Inserting Equations (5.13) and (5.6) into Equation (5.1) gives S j k gen dv > 0 (5.7) he entropy generation per unit time for the irreversible processes is indeed greater than zero since and k are positive. Note that the Joule heating and the thermal conduction in the equation are indeed irreversible. his equation satisfies the requirement for Onsager s reciprocal relations as shown in Equation (5.8). Substituting Equations (5.18) and (5.6) in Equation (5.17) yields d k j j 0 d (5.8) he homson coefficient, originally obtained from the homson relations, is written d d (5.9) In Equation (5.8), the first term is the thermal conduction, the second term is the Joule heating, and the third term is the homson heat. Note that if the Seebeck coefficient is independent of temperature, the homson coefficient is zero and then the homson heat is absent.

9 5-9 Figure 5.1 hermoelectric cooler with p-type and n-type thermoelements Numeric Solutions of homson Effect Let us consider one of p- and n-type thermoelements as shown in Figure 5.1, knowing that p- and n-type thermoelements produce the same results if the materials and dimensions are assumed to be similar. Equation (5.8) for one dimensional analysis at steady state gives the non-linear differential governing equation as d I d d dx Ak d dx I 0 A k (5.30) where A is the cross-sectional area of the thermoelement. We want to make it dimensionless. he boundary conditions will be (0) = 1 and (L) =. hen, Let 1 and 1 x L (5.31) where L is the element length. hen, the dimensionless differential equation is

10 5-10 d d d d (5.3) where the dimensionless numbers are defined as follows. he boundary condition will be that 0 0 and 1 1. Equation (5.3) was formulated on the basis of the high junction temperature to accommodate the commercial products. d I d Ak L (5.33) where is the ratio of the homson heat to the thermal conduction. Note that is not a function of. I R Ak L (5.34) where is the ratio of the Joule heating to the thermal conduction. (5.35) where is the ratio of temperature difference to the high junction temperature. he temperature difference is 1 (5.36)

11 5-11 where is the high junction temperature minus the low junction temperature 1. herefore, 1 is a function of since is fixed. he cooling power at the cold junction using Equation (5.6) is Q 1 d 1 1 I ka dx x0 (5.37) where the first term is the Peltier cooling and the second term is the thermal conduction. It has been customary in the literature for the exact solution wherein the Seebeck coefficient is evaluated at the cold junction temperature 1. he dimensionless cooling power is expressed 1 1 d d 0 (5.38) where 1 is the dimensionless cooling power, which is Q Ak 1 1 L (5.39) and the dimensionless Peltier cooling is Ak 1 I L he work per unit time gives W expressed by I I (5.40) R. hen the dimensionless work per unit time is (5.41)

12 5-1 where W Ak. hen, the coefficient of performance [11] for the thermoelectric cooler is L determined as the cooling power over the work as COP 1 (5.4) Equation (5.3) can be exactly solved for the temperature distributions conveniently with mathematical software Mathcad and then the cooling power of Equation (5.37) can be obtained, which are the exact solutions including the homson effect. Ideal Equation According to the assumption made by the both homson relations and Onsager s reciprocal relations, the thermoelectric effects and the thermal conduction in Equation (5.6) are completely independent, which implies that each term may be separately dealt with. In fact, separately dealing with the reversible processes of the three thermoelectric effects yielded the Peltier cooling of the I in the equation. However, under the assumption that the homson coefficient is negligible or the Seebeck coefficient is independent of temperature, Equation (5.30) easily provides the exact solution of the temperature distribution. hen, Equation (5.37) with the temperature distribution for the cooling power at the cold junction reduces to 1 Ak I I R Q 1 avg 1 L 1 (5.43) which appears simple but is a robust equation with a usually good agreement with experiments and with comprehensive applications in science and industry. his is here called the ideal equation. he ideal equation assumes that the Seebeck coefficient be evaluated at the average

13 5-13 temperature of [1, 13] as a result of the internal phenomena being imposed on the surface phenomena. 00 V K Intrinsic material properties Linear curve fit Linear curve fit Curve fit emperature (K) Figure 5. Seebeck coefficient as a function of temperature for Laird products Comparison between homson Effect and Ideal Equation In order to examine the homson effect on the temperature distributions with the temperature dependent Seebeck coefficient, a real commercial module of Laird CP was chosen. Both the temperature-dependent properties and the dimensions for the module were provided by the manufacturer, which are shown in able 5.1 and the temperature-dependent Seebeck coefficient is graphed in Figure 5.. able 5.1 Maximum values, Dimensions, and the Properties for a Commercial Module EC Module (Laird CP ) at 5 C max ( C) 67

14 5-14 Imax (A) 3.9 Qmax (W) 34.3 L (mm), element length 1.53 A (mm ), cross-sectional area 1 # of hermocouples 17 (), (V/K) , 50 K < < 350 K (), (V/K) , 350 K < < 450 K cmat 7 C k (W/cmK) at 7 C Module size mm o reveal the homson effect, only the Seebeck coefficients are considered to be dependent of the temperature while the electrical resistivity and the thermal conductivity are constant as shown in able 5.1. In Figure 5., the Seebeck coefficient increases with increasing the temperature up to about 350 K and then decreases with increasing the temperature, so that there is a peak value. For the solution with the non-linear differential equation, the linear curve fits for the Seebeck coefficient were used in the present work. In Figure 5., the linear curve fit of the first part for a range from 50 K to 350 K was used in the present work, while the other linear curve fit of the second part for the range from 350 K to 450 K was used separately. he linear curve fits of the two parts are shown in able 1. he dimensionless differential equation, Equation (5.3), was developed on the basis of a fixed high junction temperature so that the low junction temperature 1 decreases as increases. in Equation (5.33) is the dimensionless number indicating approximately the ratio of the homson heat to the thermal conduction. is a function only of the slope d d and the current I. Equation (5.3) with the given boundary conditions was solved for the temperature distributions conveniently using mathematical software Mathcad.

15 5-15 A typical value of = 0. was obtained for the commercial product at = 98 K and Imax = 3.9 A. And a hypothetical value of = that is fivefold larger than the typical value of 0. was used in order to closely examine the homson effect. in Equation (5.34) is the dimensionless number indicating approximately the ratio of the Joule heating to the thermal conduction. Figure 5.3 show the temperature distributions at = 98 K and I = 3.9 A as a function of along with for = 10 K ( = 15.97). he Joule heating appears dominant with a such high value of = For the commercial cases with = 0., the homson effects appear small compared to the Ideal Equation at = 0. he hypothetical cases at = 1.0 obviously provide salient examples for the homson effect. Since the temperature gradient at = 0 counteracts the Peltier cooling, the slightly lower temperature distribution near zero acts as improving the net cooling power. he homson heat is effectively described by the product of and d d. Suppose that we think of p-type thermoelement (no difference with n-type). he current flows in the positive direction. he homson heat acts as liberating heat when the d d is positive, while it acts as heat absorbing when the d d is negative. In the first half of the temperature distribution, both and d d are positive, so that the product is positive. It is known that the moving charged electrons or holes transport not only the electric energy but also the thermal energy as absorbing and liberating depending on the sign of d d along the thermoelement. he temperature distribution is slightly shifted forward giving the lower temperature distribution near = 0 which results in the improved cooling power at the cold junction. he net cooling power Q 1 in Equation (5.37) and the coefficient of performance COP [11] in Equation (5.4) along the current at = 98 K (thin lines) for = 10 K are presented in Figure 5.4. he maximum current for this commercial module is 3.9 A as shown in able 1, so the operating current barely exceeds the maximum current in practice. he slight improvement by the homson effect on Q 1 at 98K is also reflected on the corresponding COP wherein the effect appears even smaller.

16 5-16 Figure 5.3 For = 10 K ( = 15.97), dimensionless temperature vs. dimensionless distance as a function ofhe operating conditions are = 98 K and I = 3.9 A COP homson effect 10K Ideal Equation Cooling Power ( W) K Q 1 Q COP 0 98K Current (A)

17 5-17 Figure 5.4 Cooling power and COP vs. current at = 98 K and 40 K for = 10 K. = at = 40 K and = 0. at = 98 K[14] 5. hermodynamics of homson Effect he thermoelectric effect consists of three effects: the Seebeck effect, the Peltier effect, and the homson effect. Seebeck Effect he Seebeck effect is the conversion of a temperature difference into an electric current. As shown in Figure 5.5, wire A is joined at both ends to wire B and a voltmeter is inserted in wire B. Suppose that a temperature difference is imposed between two junctions, it will generally found that a potential difference or voltage V will appear on the voltmeter. he potential difference is proportional to the temperature difference. he potential difference V is V (5.44) where =H - L and is called the Seebeck coefficient (also called thermopower) which is usually measured in V/K. he sign of is positive if the electromotive force, emf, tends to drive an electric current through wire A from the hot junction to the cold junction as shown in Figure 5.5. he relative Seebeck coefficient is also expressed in terms of the absolute Seebeck coefficients of wires A and B as (5.45) A B

18 5-18 Wire A L Figure 5.5 Schematic basic thermocouple. H I _ + Wire B Wire B In practice one rarely measures the absolute Seebeck coefficient because the voltage meter always reads the relative Seebeck coefficient between wires A and B. he absolute Seebeck coefficient can be calculated from the homson coefficient. Peltier Effect When current flows across a junction between two different wires, it is found that heat must be continuously added or subtracted at the junction in order to keep its temperature constant, which is shown in Figure 5.6. he heat is proportional to the current flow and changes sign when the current is reversed. hus, the Peltier heat absorbed or liberated is Q I (5.46) Peltier where is the Peltier coefficient and the sign of is positive if the junction at which the current enters wire A is heated and the junction at which the current leaves wire A is cooled. he Peltier heating or cooling is reversible between heat and electricity. his means that heating (or cooling) will produce electricity and electricity will produce heating (or cooling) without a loss of energy.

19 5-19. Q homson,a Wire A. Q Peltier, L _ Wire B + I. Q homson,b H Wire B. Q Peltier, Figure 5.6 Schematic for the Peltier effect and the homson effect. homson Effect When current flows in wire A as shown in Figure 5.6, heat is absorbed due to the negative temperature gradient and liberated in wire B due to the positive temperature gradient, which relies on experimental observations [15, 16], depending on the materials. It is interesting to note that the wire temperature in wire B actually drops because the heat is liberated while the wire temperature in wire A rises because the heat is absorbed, which is clearly seen in the work of Amagai and Fujiki (014) [15, 16]. he homson heat is proportional to both the electric current and the temperature gradient. hus, the homson heat absorbed or liberated across a wire is Q homson = τi (5.47) where is the homson coefficient. he homson coefficient is unique among the three thermoelectric coefficients because it is the only thermoelectric coefficient directly measurable for individual materials. here is other form of heat, called Joule heating, which is irreversible and is always generated as current flows in a wire. he homson heat is reversible between heat and electricity. his heat is not the same as Joule heating, or I R.

20 5-0 homson (or Kelvin) Relationships he interrelationships between the three thermoelectric effects are important in order to understand the basic phenomena. In 1854, homson [] studied the relationships thermodynamically and provided two relationships by applying the first and second laws of thermodynamics with an assumption that the reversible and irreversible processes in thermoelectricity are separable. he necessity for the assumption remained an objection to the theory until the advent of the new thermodynamics. he relationships were later completely confirmed by experiment being essentially a consequence of Onsager s Principle [6] in It is not surprising that, since the thermal energy and electrostatic energy in quantum mechanics are often reversible, those in thermoelectrics are reversible. For a small temperature difference, the Seebeck coefficient is expressed in terms of the potential difference per unit temperature, as shown in Equation (5.44). d d (5.48) where is the electrostatic potential and is the Seebeck coefficient that is a driving force and is referred to as thermopower. Consider two dissimilar wires A and B constituting a closed circuit (Figure 5.7), in which the colder junction is at a temperature and the hotter junction is at + and both are maintained by large heat reservoirs. wo additional reservoirs are positioned at the midpoints of wires A and B. Each of these reservoirs is maintained at a temperature that is the average of those at the hotter and colder junctions.

21 5-1 Reservoir +D/ Wire A +D I Reservoir +D/ Reservoir Wire B Reservoir Figure 5.7 Closed circuit for the analysis of thermoelectric phenomena [9]. It is assumed that the thermoelectric properties such as the Seebeck, Peltier and homson coefficients remain constant with the passage of a small current. In order to study the relationships between the thermoelectric effects, a sufficiently small temperature difference is to be applied between the two junctions as shown in Figure 5.7. It is then expected that a small counter clockwise current flows (this is the way that the dissimilar wires A and B are arranged). We learned earlier that thermal energy is converted to electrical energy and that the thermal energy constitutes the Peltier heat and the homson heat, both of which are reversible. he difficulty is that the circuit inevitably involves irreversibility. he current flow and the temperature difference are always accompanied by Joule heating and thermal conduction, respectively, both of which are irreversible and make no contribution to the thermoelectric effects. homson assumed that the reversible and irreversible processes are separable. So the reversible electrical and thermal energies can be equated using the first law of thermodynamics. Electrical I I I BI AI (5.49) Energy Peltier heat at hot junction Peltier heat at cold junction homson heat in wire B homson heat in wire A For a small potential difference due to the small temperature difference, the potential difference is expressed mathematically as

22 5- d d (5.50) Using this and dividing Equation (5.49) by the current I provides d d B A (5.51) Dividing this by gives d d B A (5.5) Since is small, we conclude d d d d B A (5.53) which is nothing more than the Seebeck coefficient as defined in Equation (5.48). his is the fundamental thermodynamic theorem for a closed thermoelectric circuit; it shows the energy relationship between the electrical Seebeck effect and the thermal Peltier and homson effects. he potential difference, or in other words the electromotive force (emf), results from the Peltier heat and homson heat or vice versa. More importantly the Seebeck effect is directly caused by both to the Peltier and homson effects. he assumption of separable reversibility permits that the net change of entropy of the surroundings (reservoirs) of the closed circuit is equal to zero. he results are in excellent

23 5-3 agreement with experimental findings [17]. he net change of entropy of all reservoirs is zero because the thermoelectric effects are reversible, so we have S Q at junction Q at junction Qalong wire B Qalong wire A 0 (5.54) Using Equations (5.46) and (5.47) I I BI AI 0 (5.55) Dividing this by -I yields B A 0 (5.56) Since is small and <<, we have d d B A 0 (5.57) By taking the derivative of the first term, we have d d B A 0 (5.58)

24 5-4 which reduces to d d B A (5.59) Combining Equations (5.53) and (5.57) yields d (5.60) d Using Equation (5.48), we have a very important relationship as (5.61) Combining Equations (5.57) and (5.60) gives A B d d (5.6) Equations (5.61) and (5.6) are well known to be called the homson (Kelvin) relationships. By combining Equations (5.46) and (5.61), the thermal energy caused solely by the Seebeck effect is derived, Q I (5.63) Peltier which is one of most important thermoelectric equations and reversible thermal energy. By taking the integral of Equation (5.6) after dividing it by, the Seebeck coefficient has an expression in terms of the homson coefficients. Since the homson coefficients are measurable, the Seebeck coefficient can be calculated from them.

25 5-5 A d 0 0 B d (5.64) 5.3 Exact Solutions Equations for the Exact solutions and the Ideal Equation he ideal equation has been widely used in science and engineering. It is previously shown that the homson effect is small for commercial thermoelectric cooler modules. However, we here develop an exact solution considering the temperature-dependent material properties including the homson effect and determine the uncertainty of the ideal equation. Assuming the electric current and heat flux can be considered one-dimensional conserved quantities in a thermoelectric element as shown in Figure 5.8. he original work on this topic was discussed by Mahan (1991)[18] and later McEnaney et al. (011) [19]. q 1 EG EC L e A e x I R L I I q Figure 5.8 hermoelectric element with a load resistance or a battery.

26 5-6 From Equation (5.6), we can have d αj q = dx k (5.65) From Equation (5.17), (5.18) and (5.6), we have dq dx = αj q ρj + jα k (5.66) hese equations can be solved simultaneously numerically for a thermoelectric element in Figure 5.8 to find the temperature distribution and the heat flux profile within the element. his basic framework for determining the performance of a thermoelectric element can be incorporated into finding the efficiency of a thermoelectric generator with multiple thermoelectric elements, either a single stage, segmented, or cascaded architecture [18, 19]. Note that these equations can handle the temperature-dependent material properties. Hence, it is called the exact solution. For the solution, two boundary conditions at x = 0 and x = L are needed and either the current can be provided from Equation (5.1) for a thermoelectric generator or a constant current is given for a thermoelectric cooler. One of the difficulties in computations for the exact solution is that Equation (5.65) and (5.66) are a function of distance x while the material properties are a function of temperature, which should be resolved algebraically or numerically. Sometimes nondimensionality is helpful for a mathematical program. Now, we want to examine the ideal equation by comparing it with the exact solution to determine the uncertainty of the ideal equation. We can solve numerically for the temperature profile and the heat flux for both the ideal equation and the exact solution with the real thermoelectric material of skutterudites. he ideal equation for the temperature distribution and the heat flux profile can be obtained from Equation (5.8), which is rewritten as

27 5-7 d k j j 0 d (5.67) For the ideal equation, it is assumed that dα d = 0 which is mainly the homson effect. Solving Equation (5.67) for d dx gives d dx = ρj L k (1 x L ) h c L (5.68) he temperature distribution is obtained by integrating the above equation, which is (x) = ρj L k [x L (x L ) ] + h ( h c ) x L (5.69) From Equation (5.6) with (5.68), the heat flux as a function of the element distance x is q(x) = ρj L ( x L 1) + ( h c ) k L + αj (5.70) hese two equations for the ideal equation can be solved with the material properties at a midtemperature between h and c hermoelectric Generator In this section, the subscript 1 and denote high and low quantities, respectively, such as 1 = h and = c in Figure 5.8. he material properties for typical skutterudites [0] used in this calculation are shown in Figure 5.9.

28 5-8 ( V/K) k (W/mK) (K) (a) (b) (K) Z 0.6 (c) Figure 5.9 he curve fitting from experimental data for skutterudite by Shi et al. (011)[0], (a) Seebeckcoefficient, curve fit: α() = ( ) 10 6 V K, (b) thermal conductivity, k() = ( ) W mk, (c) electrical conductivity, ρ() = ( ) 10 6 Ωm, and (d) dimensionless figure of merit. (d) (K) he computations of the exact solution and the ideal equation for a thermoelectric generator with the temperature-dependent material properties for typical skutterudites were conducted. he high and low junction temperatures are maintained as 1 = 800 K and = 400 K, respectively. he thermoelectric element has the cross sectional area A e = 3 mm and leg length L e = mm. Both results are plotted in Figure Both the temperature distributions are very close. On the other

29 5-9 hand, the heat flow rates show a slight discrepancy between them as shown. Since we know the heat flow rate at the hot junction and cold junction temperatures from the figure, we can calculate both the thermal efficiencies using η th = q(0) q(l) q(0) (5.71) where q is the heat flow rate as a function of element distance x. he calculated thermal efficiencies for the exact solution and the ideal equation are η th = 8.6 % and η th = 9.1 %, respectively. his shows that the ideal equation with the material properties at a constant average temperature has about 6 % of uncertainty in efficiency larger than that of the exact solution for a range of temperature of 400 K 800 K. Figure 5.10 emperature profiles and heat flow rates for a thermoelectric generator at the matched load along the distance for the exact solution and the ideal equation. he following inputs are used: h = 800 K, c = 400 K, A e = 3mm, and L e = mm (this work). he material properties in Figure 5.9 were evaluated at the mid-temperature of 600 K for the ideal equation.

30 hermoelectric Coolers In this section, the subscript 1 and denote low and high quantities, respectively, such as 1 = c and = h in Figure 5.8. he material properties for bismuth telluride is shown in Figure 5.11 [1]. 30 ( V/K) k (W/mK) (a) (K) (b) (K) (c) (d) Figure 5.11 he curve fitting from experimental data for bismuth telluride [1], (a) Seebeckcoefficient, curve fit: α() = ( ) 10 6 V K, (b) thermal conductivity, k() = ( ) W mk, (c) electrical conductivity, σ() = ( ) 100 (Ωm) 1, ρ() = 1 σ() and (d) dimensionless figure of merit.

31 5-31 he computations of the exact solution and the ideal equation for a thermoelectric cooler with the temperature-dependent material properties for bismuth telluride were conducted. he high and low junction temperatures are maintained as 1 = 408 K and = 438 K, respectively. he thermoelectric element has the cross sectional area A e = 1 mm and leg length L e = 1.5 mm. he temperature and heat flow profiles for both are plotted in Figure 5.1, which show somewhat unexpected results. he temperature profile of the ideal equation appears lower than that of the exact solution, which results in the considerable lower heat flow rate. Figure 5.1 emperature profiles and heat flow rates for a thermoelectric cooler at a current of 1.3 A along the distance for the exact solution and the ideal equation. he following inputs are used: 1 = 408 K, = 438 K, A e = 1 mm, and L e = 1.5 mm (this work). he material properties for the ideal equation in Figure 5.11 were evaluated at an average temperature. he coefficient of performance COP can be calculated by COP = q(0) q(l) q(0) (5.7)

32 5-3 he calculated COPs for the exact solution and the ideal equation are COP = 1.6 and COP = 1.34, respectively. he ideal equation with the material properties at a constant average temperature underestimates about 17 % in the COP lower (underestimate) than that of the exact solution for a temperature difference of = 30 K at current of I = 1.3 A. here are a number of reports in literature about the homson effect which is only the effect of the temperature dependence of the Seebeck coefficient. However, the exact solution includes all the material properties: Seebeck coefficient, electrical resistance, and thermal conductivity. It is interesting to note that the underestimate due to the temperature dependency of material properties usually offset the losses by thermal and electrical contact resistances between thermoelements and ceramic and conductor plates. herefore, this effect has not been saliently transpired in the performance curve if they are compared with the ideal equation in thermoelectric cooler modules as usually do. We have studied the uncertainty of the ideal equation for both thermoelectric generator and cooler comparing with the exact solution. Further studies are encouraged from this starting point. 5.4 Compatibility Factor Consider the one dimensional, steady state, and thermoelectric power generation problem, where only a single element is considered. he following work is based on the work of Snyder and Ursell (003) [, 3]. Note that the coordinate x in Figure 5.13 is consistent with the previous chapters but opposite of the original work of Snyder and Ursell. Hence, care is taken that the expressions in this section may appear different from the original work, especially in sign not in quantity. he core reason of the present coordinate system is that we can use the equations developed in the previous chapters without changing the signs.

33 5-33 l A x j Figure 5.13 Diagram of a single element thermoelectric generator. he electric current density j for a simple generator is given by j = I A e (5.73) From Equation (5.13), the electric field at any position is rewritten as E j (5.74) From Equation (5.6), the heat flux (heat flow density) is rewritten as q j k (5.75) From Equation (5.18), the electrical power density is rewritten as q E j j j (5.76) From Equation (5.8), the heat balance equation is rewritten by

34 5-34 d k j j 0 d (5.77) Reduced Current Density In a thermoelectric element in Figure 5.13, the reduced current density u can be defined as the ratio of the electric current density to the heat flux by conduction.[] u j k (5.78) For a constant k, we can see that u is simply a scaled version of the current density j. Using Equations (5.77) and (5.78), we have d ( 1 u ) d dα = d ρku (5.79) Heat Balance Equation he heat balance equation in terms of the reduced current density u is finally expressed as du d = u dα d + ρku3 (5.80) which is not readily solved for u. Numerical Solution A numerical solution for Equation (5.80) was developed by Snyder [4]. Equation (5.80) can be approximated by combining the zero homson effect (dα d=0) solution and the zero resistance (ρk = 0) solution. For dα d=0 solution, the equation becomes du d = ρku3 (5.81) If we integrate both sides, we have

35 5-35 u du u u 3 c (5.8) = ρkd c which leads to 1 u = 1 1 u u c ( ρk c ) c (5.83) where ρk denotes the average of ρk between and c. For the zero resistance (ρk = 0) solution, Equation (5.80) becomes du d = u dα d (5.84) In a similar way, solving for the reduced current density, 1 u = 1 u c + (α α c ) (5.85) Combining Equation (5.83) and (5.85) gives 1 u = 1 1 u u c ( ρk c ) + (α α c ) c (5.86) In a discrete form of numerical solution for n and n 1 (across an infinitesimal layer), we have 1 = 1 1 u u n u n 1 ( ρk n n 1 ) + (α n α n 1 ) n 1 (5.87) his can be used to compute the reduced current density u() as a function of temperature with an initial u i. A specific initial u i could be obtained for a maximum efficiency using Equations (5.87) and (5.101).

36 5-36 Infinitesimal Efficiency Consider an infinitesimal layer along the distance dx. he infinitesimal efficiency η is η = q dxa e qa e = Power output Heat absorbed (5.88) Reduced Efficiency Equation (5.88) can be further developed. he total efficiency η in terms of u is η = d u(α + ρku) αu 1 (5.89) he d is recognizable as the infinitesimal Carnot efficiency. Reduced Efficiency So we define the reduced efficiency η r as η c = d = = h c = 1 c (5.90) h h η = η c η r (5.91) where u(α + ρku) η r = αu 1 (5.9) or, equivalently, if α and u are not zero, we have an expression as

37 5-37 η r = 1 + u α Z 1 1 αu (5.93) Compatibility Factor It is found in Equation (5.93) that there is a largest reduced efficiency between u = 0 and u = Z α. Hence, taking derivative of the reduced efficiency η r with respect to the reduced current density u and setting it to zero gives a specific reduced current u that is called the compatibility factor s. s = Z α (5.94) Replacing this in place of u in Equation (5.93) gives the maximum reduced efficiency η r,max as 1 + Z 1 η r,max = 1 + Z + 1 (5.95) As a general rule, η r () is significantly compromised when u deviated from s by more than a factor of two []. Segmented hermoelements On the applications of large temperature differences such as solar thermoelectric generators, the segmented thermoelement, which consists of more than one material in a thermoelectric leg, is often considered because one material cannot have good efficiency or sustainability over the temperature range. he compatibility factor is then a measure of maximum suitability over the segmented materials since one current should flow through the segmented thermoelement. he compatibility factor is a thermodynamic properties essential for designing an efficient segmented

38 5-38 thermoelectric device. If the compatibility factors differ by a factor of or more, the maximum efficiency can in fact decrease by segmentation []. able 5. Spread sheet Calculation of p-type Element Performance based on Snyder and Ursell (003) [, 4]. (K) Material α 10-3 k Z u (1/V) s (1/V) η r (%) η (%) (V/K) (cm) (W/mK) 973 CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb CeFe 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Zn 4Sb Bi e Bi e Bi e Bi e Bi e Bi e Bi e

39 Bi e Bi e hree potential materials, Bie3, Zn4Sb3, and CeFe4Sb3 as a segmented element are examined for adequacy. he three thermoelectric material properties along with temperature are listed in able 5.. he reduced current density u, compatibility factors s, reduced efficiency η r and maximum reduced efficiency η r,max are computed numerically as shown in the table using Equations (5.87), (5.94), (5.9), and (5.95) with an initial u i = that has to be a value corresponding to the maximum efficiency. he computed results are plotted in Figure he reduced current density u, which was defined as the ratio of the current density j to the heat flow density k, is a numerical solution from the heat balance equation of Equation (5.80). he reduced current density remains almost constant over the temperature. However, the compatibility factor s, which is a value corresponding to the maximum reduced efficiency, shows the maximum potential of the material. So the discrepancy between u and s causes the decrease of the reduced efficiency, which is shown in Figure 5.14 (b). his should be a good measure of selecting segmented materials.

40 5-40 (a) (b) Figure 5.14 (a) Variation of reduced current density with temperature for a typical thermoelectric generator. (b) Reduced efficiency compared to the maximum reduced efficiency. hese plots are based on the work of Snyder (006) [, 4].

41 5-41 hermoelectric Potential From Equation (5.75), the thermoelectric potential Φ is defined as Φ = q j = α 1 u (5.96) hen, q = jφ (5.97) When we take derivative of Equation (5.96), using Equation (5.74) we find E = Φ (5.98) From Equation (5.76), the electrical power density is expressed as q = Ej = j Φ (5.99) From Equation (5.88), the infinitesimal efficiency η is developed as η = q dx q Hence, the infinitesimal efficiency is = j Φdx dφ jφ = dx dx Φ = Φ Φ = Φ h Φ c = 1 Φ c Φ h Φ h η = 1 Φ α c c 1 c u = 1 c Φ h α h h 1 u h (5.100) (5.101)

42 5-4 Using Equation (5.93) and (5.96), we have η r = dlnφ dt (5.10) aking integration of Equation (5.10) gives Φ h Φ c = exp [ h η r d ] c (5.103) From Equation (5.101), the overall efficiency of a finite segment is given by η = 1 exp [ h η r d ] c (5.104) Now we want to obtain the total efficiency when, and k are constant with respect to temperature, the performance of a generator operating at maximum efficiency can be calculated analytically. he resulting maximum efficiency is given by η max = h c h 1 + Z Z + c h (5.105) his equation is the normally attained maximum conversion efficiency in Chapter with assumption of the constant material properties.

43 5-43 Problems 5.1. Derive Equation (5.6) and (5.8) in detail. 5.. Describe the homson heat What is the homson relations? 5.4. Derive Equations (5.65) and (5.66) of the exact solutions and explain what is the meaning of the exact solution Derive Equations (5.69) and (5.70) of the ideal equation and explain what is the meaning of the ideal equation Derive in detail Equation (5.94) Derive in detail Equation (5.95). Projects 5.8. (Half term project) Develop a Mathcad program to provide Figure 5.10 for a thermoelectric generator and explore the uncertainty of the ideal equation comparing with the exact solution (half term project) Develop a Mathcad program to provide Figure 5.1 for a thermoelectric cooler and explore the uncertainty of the ideal equation comparing with the exact solution.

44 5-44 References 1. Peltier, J.C., New experiments on the calorific electric currents. Annales de chimie de physique, (): p homson, W., Account of researches in thermo-electricity. Proceedings of the Royal Society of London, : p Callen, H.B., he Application of Onsager's Reciprocal Relations to hermoelectric, hermomagnetic, and Galvanomagnetic Effects. Physical Review, (11): p Casimir, H.B.G., On Onsager's Principle of Microscopic Reversibility. Reviews of Modern Physics, (-3): p Verhas, J., Onsager's reciprocal relations and some basic laws. Journal of Computational and Applied Mechanics, (1): p Onsager, L., Reciprocal Relations in Irreversible Processes. I. Physical Review, (4): p Landau, L.D. and E.M. Lifshitz, Electrodynamics of continuous media. 1960: Pergamon Press, Oxford, UK. 8. Ioffe, A.F., Semiconductor thermoelement and thermoelectric cooling. 1957, London: Infosearch Limited Rowe, D.M., CRC handbook of thermoelectrics. 1995, Boca Raton London New York: CRC Press. 10. Lee, H., hermal Design; Heat Sink, hermoelectrics, Heat Pipes, Compact Heat Exchangers, and Solar Cells. 010, Hoboken, New Jersey: John Wiley & Sons. 11. Xie, W., et al., Identifying the specific nanostructures responsible for the high thermoelectric performance of (Bi,Sb)e3 nanocomposites. Nano Lett, (9): p Goldsmid, H.J., Introduction to thermoelectricity. 010, Heidelberg: Springer. 13. Nolas, G.S., J. Sharp, and H.J. Goldsmid, hermoelectrics: basic principles and new materials developments. 001, Berlin: Springer,. 14. Lee, H., he homson effect and the ideal equation on thermoelectric coolers. Energy, : p Nettleton, H.R. he homson Effect. in Proceedings of the Physical Society of London Amagai, Y. and H. Fujiki, Measurement of the homson heat distribution in a thin-wire metal, in Precision Electromagnetic Measurements (CPEM 014) p Benedict, R.P., Fundamentals of emperature, Pressure, and Flow Measurements, ed. r. Ed. 1984, New York: Wiley-Interscience. 18. Mahan, G.D., Inhomogeneous thermoelectrics. Journal of Applied Physics, (8): p McEnaney, K., et al., Modeling of concentrating solar thermoelectric generators. Journal of Applied Physics, (7): p

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