5-1
Chapter 5 Thomson Effect, Exact Solution, and Compatibility Factor
The formulation of the classical basic equations for a thermoelectric cooler from the Thomson relations to the non-linear differential equation with Onsager’s reciprocal relations was performed to basically study the Thomson effect in conjunction with the ideal equation. The ideal equation is obtained when the Thomson coefficient is assumed to be zero. The exact solutions derived for a commercial thermoelectric cooler module provided the temperature distributions including the Thomson effect. The positive Thomson coefficient led to a slight improvement on the performance of the thermoelectric device while the negative Thomson coefficient led to a slight declination of the performance. The 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. The Thomson effect is small for typical commercial thermoelectric coolers and the ideal equation effectively predicts the performance.
5.1 Thomson Effects
Thermoelectric phenomena are useful and have drawn much attention since the discovery of the phenomena in the early nineteenth century. The barriers to applications were low efficiencies and 5-2
the availability of materials. In 1821, Thomas 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. This 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]. This is called the Peltier effect (or Peltier cooling). In 1854, William Thomson 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 [2]. This is called the Thomson effect (or Thomson heat). These three effects are called the thermoelectric effects. Thomson 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 [2]. The relationship developed is called the Thomson (or Kelvin) relations. The necessity for the assumption remained an objection to the theory until the advent of the new thermodynamics [3-5]. The 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 Thomson relations provide a simple expression for Peltier cooling, which is the product of the Seebeck coefficient, the temperature at the junction, and the current. This Peltier cooling is the principal thermoelectric cooling mechanism. There are two counteracting phenomena, which are the Joule heating and the thermal conduction. The 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]
5-3
5.1.1 Formulation of Basic Equations
Onsager’s Reciprocal Relations The second law of thermodynamics with no mass transfer (S = 0) in an isotropic substance provides an expression for the entropy generation.
2 Q (5.1) S gen 1 T
Suppose that the derivatives of the entropy generation sgen per unit volume with respect to arbitrary quantities xi are quantities Ji [4, 8] as
sgen (5.2) J i xi
At maximum sgen, Ji is zero. Accordingly, at the state close to equilibrium (or maximum), Ji is small [3]. The entropy generation per unit time and unit volume is
s x (5.3) s gen i J gen t t i
We want to express xi t as X i . Then, we have
sgen X i Ji (5.4)
The X i are usually a function of J i . Onsager [6] stated that if were completely independent, we should have the relation, expanding X i in the powers of Ji and taking only linear terms [7]. The smallness of in practice allows the linear relations being sufficient.
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(5.5) X i Rij J j j where Rij are called the kinetic coefficients. Hence, we have
(5.6) sgen X i Ji i
It is necessary to choose the quantities X i in some manner, and then to determine the Ji. The X i and Ji are conveniently determined simply by means of the formula for the rate of change of the total entropy generation.
S X J dV > 0 (5.7) gen i i i
For two terms, we have
S X J X J dV > 0 (5.8) gen 1 1 2 2
And
X1 R11 J1 R12 J 2 (5.9)
X 2 R21J1 R22 J2 (5.10)
From these equations, one can assert that the kinetic coefficients are symmetrical with respect to the suffixes 1 and 2.
R12 R21 (5.11) which is called the reciprocal relations. 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 (5.12)
The electric field E is affected by the current density j and the temperature gradientT . The coefficients are known according to the Ohm’s law and the Seebeck effect [7]. The field is then expressed as
E j T (5.13) where is the electrical resistivity and is the Seebeck coefficient. Solving for the current density gives
1 (5.14) j E T
The heat flux q is also affected by the both field E and temperature gradient . However, the coefficients were not readily attainable at that time. Thomson 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 Thomson relationship using Onsager’s principle. The heat flux with arbitrary coefficients is
(5.15) q C1E C2T
The general heat diffusion equation is 5-6
T (5.16) q q c p t
For steady state, we have q q 0 (5.17) where q is expressed by [7] q E j j 2 j T (5.18)
Expressing Equation (5.1) with Equation (5.17) in a manner that the heat flux gradient q has a minus sign, the entropy generation per unit time will be
2 Q 1 (5.19) S q q dV gen 1 T T
Using Equation (5.18), we have
E j q (5.20) S dV gen T T
The 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]. Then, we have
E j q T (5.21) S dV gen 2 T T 5-7
If we take j and q as X1 and X 2 , then the corresponding quantities J1 and J 2 are the components of the vectors E T and T T 2 [7]. Accordingly, as shown in Equations (5.9) and (5.10) with Equation (5.15), we have
1 E 1 T (5.22) j T T 2 2 T T
E T (5.23) q C T C T 2 1 2 2 T T where the reciprocal relations in Equation (5.11) gives
1 (5.24) C T T 2 1
Thus, we have C1 T . Inserting this and Equation (5.13) into (5.23) gives
2T (5.25) q Tj C2 T
We introduce the Wiedemann-Franz law that is Lo k T , where Lo is the Lorentz number
2 (constant). Expressing k as LoT/, we find that the coefficient C2 T is nothing more than the thermal conductivity k. Finally, the heat flow density vector (heat flux) is expressed as
q Tj kT (5.26)
This is the most essential equation for thermoelectric phenomena. The second term pertains to the thermal conduction and the term of interest is the first term, which gives the thermoelectric 5-8
effects: directly the Peltier cooling but indirectly the Seebeck effect and the Thomson heat [8- 10]. Inserting Equations (5.13) and (5.26) into Equation (5.21) gives
2 2 (5.27) j k T S dV > 0 gen 2 T T
The 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. This equation satisfies the requirement for Onsager’s reciprocal relations as shown in Equation (5.8). Substituting Equations (5.18) and (5.26) in Equation (5.17) yields
d (5.28) kT j 2 T j T 0 dT
The Thomson coefficient , originally obtained from the Thomson relations, is written
d (5.29) T dT
In Equation (5.28), the first term is the thermal conduction, the second term is the Joule heating, and the third term is the Thomson heat. Note that if the Seebeck coefficient is independent of temperature, the Thomson coefficient is zero and then the Thomson heat is absent.
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Figure 5.1 Thermoelectric cooler with p-type and n-type thermoelements.
5.1.2 Numeric Solutions of Thomson 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 (5.28) for one dimensional analysis at steady state gives the non-linear differential governing equation as d (5.30) I d 2T dT I 2 dT T 0 dx2 Ak dx A2k where A is the cross-sectional area of the thermoelement. We want to make it dimensionless. The boundary conditions will be T(0) = T1 and T(L) = T2. Then, Let
T T x (5.31) 1 and T2 T1 L where L is the element length. Then, the dimensionless differential equation is
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d 2 d (5.32) 1 1 0 d 2 d where the dimensionless numbers are defined as follows. The boundary condition will be that 0 0 and 1 1. Equation (5.32) was formulated on the basis of the high junction temperature T2 to accommodate the commercial products.
d (5.33) IT2 T dT T Ak L where is the ratio of the Thomson heat to the thermal conduction. Note that is not a function of T.
I 2 R (5.34) T Ak L where is the ratio of the Joule heating to the thermal conduction.
T (5.35) T2 where is the ratio of temperature difference to the high junction temperature. The temperature difference is
T T2 T1 (5.36) 5-11
where is the high junction temperature T2 minus the low junction temperature T1. Therefore,
T1 is a function of since T2 is fixed. The cooling power at the cold junction using Equation (5.26) is
dT (5.37) Q1 T1 T1I kA dx x0 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 T1. The dimensionless cooling power is expressed
1 d (5.38) 1 d 0 where 1 is the dimensionless cooling power, which is
Q (5.39) 1 1 AkT L and the dimensionless Peltier cooling is
T I (5.40) 1 Ak L
The work per unit time gives W IT I 2 R . Then the dimensionless work per unit time is expressed by
(5.41) 5-12
W where . Then, the coefficient of performance [11] for the thermoelectric cooler is AkT L determined as the cooling power over the work as
(5.42) COP 1
Equation (5.32) 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 Thomson effect.
Ideal Equation According to the assumption made by the both Thomson relations and Onsager’s reciprocal relations, the thermoelectric effects and the thermal conduction in Equation (5.26) 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 TI in the equation. However, under the assumption that the Thomson coefficient is negligible or the Seebeck coefficient is independent of temperature, Equation (5.30) easily provides the exact solution of the temperature distribution. Then, Equation (5.37) with the temperature distribution for the cooling power at the cold junction reduces to
1 Ak (5.43) Q T T I I 2 R T T 1 avg 1 2 L 2 1
which appears simple but is a robust equation with a usually good agreement with experiments and with comprehensive applications in science and industry. This is here called the ideal equation. The ideal equation assumes that the Seebeck coefficient be evaluated at the average 5-13
temperature of T [12, 13] as a result of the internal phenomena being imposed on the surface phenomena.
200
V 180 K Intrinsic material properties Linear curve fit 160 Linear curve fit Curve fit 140 200 250 300 350 400 450 500
Temperature (K)
Figure 5.2 Seebeck coefficient as a function of temperature for Laird products.
5.1.3 Comparison between Thomson Effect and Ideal Equation
In order to examine the Thomson effect on the temperature distributions with the temperature dependent Seebeck coefficient, a real commercial module of Laird CP10-127-05 was chosen. Both the temperature-dependent properties and the dimensions for the module were provided by the manufacturer, which are shown in Table 5.1 and the temperature-dependent Seebeck coefficient is graphed in Figure 5.2.
Table 5.1 Maximum values, Dimensions, and the Properties for a Commercial Module TEC Module (Laird CP10-127-05) at 25°C
Tmax (°C) 67 5-14
Imax (A) 3.9
Qmax (W) 34.3 L (mm), element length 1.253 A (mm2), cross-sectional area 1 # of Thermocouples 127 (T), (V/K) 0.2068T+138.78, 250 K < T< 350 K (T), (V/K) −0.2144T+281.02, 350 K < T < 450 K cmat 27 °C 1.01 × 10-3 k (W/cmK) at 27 °C 1.51 × 10-2 Module size 30 × 30 × 3.2 mm
To reveal the Thomson 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 Table 5.1. In Figure 5.2, 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 5.2, the linear curve fit of the first part for a range from 250 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. The linear curve fits of the two parts are shown in Table 1. The dimensionless differential equation, Equation (5.32), was developed on the basis of a fixed high junction temperature T2 so that the low junction temperature T1 decreases as T increases. in Equation (5.33) is the dimensionless number indicating approximately the ratio of the Thomson heat to the thermal conduction. is a function only of the slope d dT and the current I. Equation (5.32) with the given boundary conditions was solved for the temperature distributions conveniently using mathematical software Mathcad. 5-15
A typical value of = 0.2 was obtained for the commercial product at T2 = 298 K and
Imax = 3.9 A. And a hypothetical value of = that is fivefold larger than the typical value of 0.2 was used in order to closely examine the Thomson 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 T2 = 298 K and I = 3.9 A as a function of along with for T = 10 K ( = 15.97). The Joule heating appears dominant with a such high value of = 15.97. For the commercial cases with = 0.2, the Thomson effects appear small compared to the Ideal Equation at = 0. The hypothetical cases at = 1.0 obviously provide salient examples for the Thomson 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. The Thomson heat is effectively described by the product of and d d . Suppose that we think of p-type thermoelement (no difference with n-type). The current flows in the positive direction. The Thomson 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 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 along the thermoelement. The temperature distribution is slightly shifted forward giving the lower temperature distribution near = 0 which results in the improved cooling power at the cold junction. The net cooling power Q1 in Equation (5.37) and the coefficient of performance COP
[11] in Equation (5.42) along the current at T2 = 298 K (thin lines) for T = 10 K are presented in Figure 5.4. The maximum current for this commercial module is 3.9 A as shown in Table 1, so the operating current barely exceeds the maximum current in practice. The slight improvement by the Thomson effect on Q1 at 298K is also reflected on the corresponding COP wherein the effect appears even smaller. 5-16
Figure 5.3 For T = 10 K ( = 15.97), dimensionless temperature vs. dimensionless distance
as a function ofThe operating conditions are T2 = 298 K and I = 3.9 A.
70 4 Thomson effect T 10K 60 Ideal Equation 3.5 COP 50 Q1 3 T2 420K 40 2.5
30 2
COP Q1 20 1.5
Cooling Power ( W) (Cooling Power 10 1 T 298K 0 2 0.5
10 0 0 1 2 3 4 5 6 Current (A)
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Figure 5.4 Cooling power and COP vs. current at T2 = 298 K and 420 K for T = 10 K. = -
0.29 at T2 = 420 K and = 0.2 at T2 = 298 K[14]
5.2 Thermodynamics of Thomson Effect
The thermoelectric effect consists of three effects: the Seebeck effect, the Peltier effect, and the Thomson effect.
Seebeck Effect The 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. The potential difference is proportional to the temperature difference. The potential difference V is
V AB T (5.44)
where T=TH - TL and is called the Seebeck coefficient (also called thermopower) which is usually measured in V/K. The 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. The relative Seebeck coefficient is also expressed in terms of the absolute Seebeck coefficients of wires A and B as
AB A B (5.45)
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Wire A
T TH L I _ + Wire B Wire B Figure 5.5 Schematic basic thermocouple.
In practice one rarely measures the absolute Seebeck coefficient because the voltage meter always reads the relative Seebeck coefficient between wires A and B. The absolute Seebeck coefficient can be calculated from the Thomson 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. The heat is proportional to the current flow and changes sign when the current is reversed. Thus, the Peltier heat absorbed or liberated is
(5.46) QPeltier ABI
where AB is the Peltier coefficient and the sign of AB 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. The Peltier heating or cooling is reversible between heat and electricity. This means that heating (or cooling) will produce electricity and electricity will produce heating (or cooling) without a loss of energy.
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. QThomson,A Wire A
. TL TH . Q Q Peltier,AB I Peltier,AB _ Wire B + Wire B . Q Thomson,B Figure 5.6 Schematic for the Peltier effect and the Thomson effect.
Thomson 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 (2014) [15, 16]. The Thomson heat is proportional to both the electric current and the temperature gradient. Thus, the Thomson heat absorbed or liberated across a wire is
푄̇ 푇ℎ표푚푠표푛 = −휏퐼∇푇 (5.47) where is the Thomson coefficient. The Thomson coefficient is unique among the three thermoelectric coefficients because it is the only thermoelectric coefficient directly measurable for individual materials. There is other form of heat, called Joule heating, which is irreversible and is always generated as current flows in a wire. The Thomson heat is reversible between heat and electricity. This heat is not the same as Joule heating, or I2R.
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Thomson (or Kelvin) Relationships The interrelationships between the three thermoelectric effects are important in order to understand the basic phenomena. In 1854, Thomson [2] 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. The necessity for the assumption remained an objection to the theory until the advent of the new thermodynamics. The relationships were later completely confirmed by experiment being essentially a consequence of Onsager’s Principle [6] in 1931. 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 (5.48) T dT AB 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 T and the hotter junction is at T+T and both are maintained by large heat reservoirs. Two 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.
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Reservoir T+DT/2 Wire A
T T+DT I
Reservoir Reservoir T+DT/2 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 Thomson 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 Thomson heat, both of which are reversible. The difficulty is that the circuit inevitably involves irreversibility. The 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. Thomson 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.
I I T T I T IT IT (5.49) AB AB B A Electrical Energy Peltier heat Peltier heat Thomson heat Thomson heat at hot junction at cold junction in wire B in wire A
For a small potential difference due to the small temperature difference, the potential difference is expressed mathematically as 5-22
d (5.50) T dT
Using this and dividing Equation (5.49) by the current I provides
d (5.51) T T T T T dT AB AB B A
Dividing this by T gives
d T T T (5.52) AB AB dT T B A
Since T is small, we conclude
d d (5.53) AB dT dT B A which is nothing more than the Seebeck coefficient as defined in Equation (5.48). This 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 Thomson effects. The potential difference, or in other words the electromotive force (emf), results from the Peltier heat and Thomson heat or vice versa. More importantly the Seebeck effect is directly caused by both to the Peltier and Thomson effects. The assumption of separable reversibility permits that the net change of entropy of the surroundings (reservoirs) of the closed circuit is equal to zero. The results are in excellent 5-23
agreement with experimental findings [17]. The net change of entropy of all reservoirs is zero because the thermoelectric effects are reversible, so we have
Q Q Q Q (5.54) S at junction TT at junction T along wire B along wire A 0 T T T T T 2 T T 2
Using Equations (5.46) and (5.47)
I T T I T IT IT (5.55) AB AB B A 0 T T T T T 2 T T 2
Dividing this by -IT yields
T T T (5.56) AB AB T T T B A 0 T T T 2 T T 2
Since T is small and T 2 << T, we have
(5.57) d AB T B A 0 dT T
By taking the derivative of the first term, we have
d (5.58) T AB AB dT B A 0 T 2 T
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which reduces to
d (5.59) AB AB T dT B A
Combining Equations (5.53) and (5.57) yields
d (5.60) AB dT T
Using Equation (5.48), we have a very important relationship as
AB ABT (5.61)
Combining Equations (5.57) and (5.60) gives
d (5.62) T AB A B dT
Equations (5.61) and (5.62) are well known to be called the Thomson (Kelvin) relationships. By combining Equations (5.46) and (5.61), the thermal energy caused solely by the Seebeck effect is derived,
(5.63) QPeltier ABTI which is one of most important thermoelectric equations and reversible thermal energy. By taking the integral of Equation (5.62) after dividing it by T, the Seebeck coefficient has an expression in terms of the Thomson coefficients. Since the Thomson coefficients are measurable, the Seebeck coefficient can be calculated from them. 5-25
T T (5.64) A dT B dT AB 0 T 0 T
5.3 Exact Solutions
5.3.1 Equations for the Exact solutions and the Ideal Equation
The ideal equation has been widely used in science and engineering. It is previously shown that the Thomson effect is small for commercial thermoelectric cooler modules. However, we here develop an exact solution considering the temperature-dependent material properties including the Thomson 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. The original work on this topic was discussed by Mahan (1991)[18] and later McEnaney et al. (2011) [19].
q1 TEG TEC T1 - - - Ae - x - R Le L - I I I
T2
q 2 Figure 5.8 Thermoelectric element with a load resistance or a battery. 5-26
From Equation (5.26), we can have
푑푇 훼푇푗 − 푞 (5.65) = 푑푥 푘
From Equation (5.17), (5.18) and (5.26), we have
푑푞 훼푇푗 − 푞 (5.66) = 𝜌푗2 + 푗훼 푑푥 푘
These 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. This 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 (5.21) 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 non- dimensionality 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. The ideal equation for the temperature distribution and the heat flux profile can be obtained from Equation (5.28), which is rewritten as
5-27
d (5.67) kT j 2 T j T 0 dT
For the ideal equation, it is assumed that 푑훼⁄푑푇 = 0 which is mainly the Thomson effect. Solving
Equation (5.67) for 푑푇⁄푑푥 gives
푑푇 𝜌푗2퐿 푥 푇 − 푇 (5.68) = (1 − ) − ℎ 푐 푑푥 푘 퐿 퐿
The temperature distribution is obtained by integrating the above equation, which is
𝜌푗2퐿2 푥 푥 2 푥 (5.69) 푇(푥) = [ − ( ) ] + 푇 − (푇 − 푇 ) 2푘 퐿 퐿 ℎ ℎ 푐 퐿
From Equation (5.26) with (5.68), the heat flux as a function of the element distance x is
푥 푘 (5.70) 푞(푥) = 𝜌푗2퐿 ( − 1) + (푇 − 푇 ) + 훼푇푗 퐿 ℎ 푐 퐿
These two equations for the ideal equation can be solved with the material properties at a mid- temperature between 푇ℎ and 푇푐.
5.3.2 Thermoelectric Generator
In this section, the subscript 1 and 2 denote high and low quantities, respectively, such as 푇1 =
푇ℎ and 푇2 = 푇푐 in Figure 5.8. The material properties for typical skutterudites [20] used in this calculation are shown in Figure 5.9.
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200 2.8 180 2.6
V/K) 160 2.4
(
k (W/mK) 140 2.2
120 2 300 400 500 600 700 800 300 400 500 600 700 800 T (K) T (K) (a) (b)
1
0.8
0.6
ZT
0.4
0.2 300 400 500 600 700 800 T (K) (c) (d) Figure 5.9 The curve fitting from experimental data for skutterudite by Shi et al. (2011)[20], (a) Seebeckcoefficient, curve fit: 훼(푇) = (60.703 + 0.2482푇 − 0.0001푇2) × 10−6 푉⁄퐾, (b) thermal conductivity, 푘(푇) = (2.388 + 0.0025푇 − 6 × 10−6푇2 + 4 × 10−9푇3) 푊⁄푚퐾, (c) electrical conductivity, 𝜌(푇) = (1.1527 + 0.0111푇 + 4 × 10−6푇2) × 10−6Ω푚, and (d) dimensionless figure of merit.
The computations of the exact solution and the ideal equation for a thermoelectric generator with the temperature-dependent material properties for typical skutterudites were conducted. The high and low junction temperatures are maintained as 푇1 = 800 K and 푇2 = 400 K, respectively. The 2 thermoelectric element has the cross sectional area 퐴푒 = 3 mm and leg length 퐿푒 = 2 mm. Both results are plotted in Figure 5.10. Both the temperature distributions are very close. On the other 5-29
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
푞(0) − 푞(퐿) (5.71) 휂 = 푡ℎ 푞(0) where 푞 is the heat flow rate as a function of element distance x. The calculated thermal efficiencies for the exact solution and the ideal equation are 휂푡ℎ = 8.6 % and 휂푡ℎ = 9.1 %, respectively. This 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 Temperature profiles and heat flow rates for a thermoelectric generator at the matched load along the distance for the exact solution and the ideal equation. The following 2 inputs are used: 푇ℎ = 800 K, 푇푐 = 400 K, 퐴푒 = 3푚푚 , and 퐿푒 = 2푚푚 (this work). The material properties in Figure 5.9 were evaluated at the mid-temperature of 600 K for the ideal equation.
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5.3.3 Thermoelectric Coolers
In this section, the subscript 1 and 2 denote low and high quantities, respectively, such as 푇1 =
푇푐 and 푇2 = 푇ℎ in Figure 5.8. The material properties for bismuth telluride is shown in Figure 5.11 [21].
230
220 1
V/K) 210
( 0.5
k (W/mK) 200
190 0 300 350 400 450 500 300 350 400 450 500 T (K) T (K) (a) (b)
(c) (d) Figure 5.11 The curve fitting from experimental data for bismuth telluride [21], (a) Seebeckcoefficient, curve fit: 훼(푇) = (−103.02 + 1.4882푇 − 0.0017푇2) × 10−6 푉⁄퐾, (b) thermal conductivity, 푘(푇) = (3.2425 − 0.017푇 + 1.5 × 10−5푇2) 푊⁄푚퐾, (c) electrical conductivity, 𝜎(푇) = (4105.5 − 13.211푇 + 0.0121 푇2) × 100 (Ω푚)−1, 𝜌(푇) = 1⁄𝜎(푇) and (d) dimensionless figure of merit.
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The computations of the exact solution and the ideal equation for a thermoelectric cooler with the temperature-dependent material properties for bismuth telluride were conducted. The high and low junction temperatures are maintained as 푇1 = 408 K and 푇2 = 438 K, respectively. The 2 thermoelectric element has the cross sectional area 퐴푒 = 1 mm and leg length 퐿푒 = 1.5 mm. The temperature and heat flow profiles for both are plotted in Figure 5.12, which show somewhat unexpected results. The temperature profile of the ideal equation appears lower than that of the exact solution, which results in the considerable lower heat flow rate.
Figure 5.12 Temperature 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. The following inputs are 2 used: 푇1 = 408 K, 푇2 = 438 K, 퐴푒 = 1 푚푚 , and 퐿푒 = 1.5 푚푚 (this work). The material properties for the ideal equation in Figure 5.11 were evaluated at an average temperature.
The coefficient of performance COP can be calculated by
푞(0) (5.72) 퐶푂푃 = 푞(퐿) − 푞(0) 5-32
The calculated COPs for the exact solution and the ideal equation are COP = 1.6 and COP = 1.34, respectively. The 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 ∆푇 = 30 K at current of I = 1.3 A. There are a number of reports in literature about the Thomson 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. Therefore, 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. The following work is based on the work of Snyder and Ursell (2003) [22, 23]. Note that the coordinate 푥 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. The core reason of the present coordinate system is that we can use the equations developed in the previous chapters without changing the signs.
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T1 - - - A - x - l -
j
T 2 Figure 5.13 Diagram of a single element thermoelectric generator.
The electric current density 푗 for a simple generator is given by
퐼 (5.73) 푗 = 퐴푒
From Equation (5.13), the electric field at any position is rewritten as
E j T (5.74)
From Equation (5.26), the heat flux (heat flow density) is rewritten as
q Tj kT (5.75)
From Equation (5.18), the electrical power density is rewritten as
q E j j 2 j T (5.76)
From Equation (5.28), the heat balance equation is rewritten by 5-34
d (5.77) kT j 2 T j T 0 dT
Reduced Current Density In a thermoelectric element in Figure 5.13, the reduced current density 푢 can be defined as the ratio of the electric current density to the heat flux by conduction.[22] 푗 (5.78) 푢 ≡ 푘∇푇
For a constant 푘∇푇, we can see that 푢 is simply a scaled version of the current density 푗. Using Equations (5.77) and (5.78), we have
1 (5.79) 푑 ( ) 푑훼 푢 = 푇 − 𝜌푘푢 푑푇 푑푇
Heat Balance Equation The heat balance equation in terms of the reduced current density 푢 is finally expressed as
푑푢 푑훼 (5.80) = −푢2푇 + 𝜌푘푢3 푑푇 푑푇 which is not readily solved for 푢.
Numerical Solution A numerical solution for Equation (5.80) was developed by Snyder [24]. Equation (5.80) can be approximated by combining the zero Thomson effect (푑훼⁄푑푇=0) solution and the zero resistance (𝜌푘 = 0) solution. For 푑훼⁄푑푇=0 solution, the equation becomes
푑푢 (5.81) = 𝜌푘푢3 푑푇
If we integrate both sides, we have 5-35
푢 푑푢 푇 (5.82) ∫ = ∫ 𝜌푘푑푇 푢3 푢푐 푇푐 which leads to
1 1 (5.83) 2̅̅̅̅ = √1 − 2푢푐 𝜌푘(푇 − 푇푐) 푢 푢푐
̅̅̅̅ where 𝜌푘 denotes the average of 𝜌푘 between 푇 and 푇푐. For the zero resistance (𝜌푘 = 0) solution, Equation (5.80) becomes
푑푢 푑훼 (5.84) = −푢2푇 푑푇 푑푇
In a similar way, solving for the reduced current density,
1 1 (5.85) = + 푇̅(훼 − 훼푐) 푢 푢푐
Combining Equation (5.83) and (5.85) gives
1 1 (5.86) 2̅̅̅̅ = √1 − 2푢푐 𝜌푘(푇 − 푇푐) + 푇̅(훼 − 훼푐) 푢 푢푐
In a discrete form of numerical solution for 푛 and 푛 − 1 (across an infinitesimal layer), we have
1 1 (5.87) 2 ̅̅̅̅ ̅ = √1 − 2푢푛−1𝜌푘(푇푛 − 푇푛−1) + 푇(훼푛 − 훼푛−1) 푢푛 푢푛−1
This can be used to compute the reduced current density 푢(푇) as a function of temperature with an initial 푢𝑖. A specific initial 푢𝑖 could be obtained for a maximum efficiency using Equations (5.87) and (5.101). 5-36
Infinitesimal Efficiency Consider an infinitesimal layer along the distance dx. The infinitesimal efficiency 휂 is
푞̇푑푥퐴 푃표푤푒푟 표푢푡푝푢푡 (5.88) 휂 = 푒 = 푞퐴푒 퐻푒푎푡 푎푏푠표푟푏푒푑
Reduced Efficiency Equation (5.88) can be further developed. The total efficiency 휂 in terms of 푢 is
푑푇 푢(훼 + 𝜌푘푢) (5.89) 휂 = 푇 1 훼푢 − 푇
The 푑푇⁄푇 is recognizable as the infinitesimal Carnot efficiency.
푑푇 ∆푇 푇ℎ − 푇푐 푇푐 (5.90) 휂푐 = = = = 1 − 푇 푇 푇ℎ 푇ℎ
Reduced Efficiency
So we define the reduced efficiency 휂푟 as
휂 = 휂푐휂푟 (5.91) where 푢(훼 + 𝜌푘푢) (5.92) 휂 = 푟 1 훼푢 − 푇 or, equivalently, if 훼 and 푢 are not zero, we have an expression as
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훼 1 + 푢 (5.93) 휂 = 푍 푟 1 1 − 훼푢푇
Compatibility Factor It is found in Equation (5.93) that there is a largest reduced efficiency between 푢 = 0 and 푢 =
푍⁄훼. Hence, taking derivative of the reduced efficiency 휂푟 with respect to the reduced current density 푢 and setting it to zero gives a specific reduced current 푢 that is called the compatibility factor 푠.
1 − √1 + 푍푇 (5.94) 푠 = 훼푇
Replacing this in place of 푢 in Equation (5.93) gives the maximum reduced efficiency 휂푟,푚푎푥 as
√1 + 푍푇 − 1 (5.95) 휂푟,푚푎푥 = √1 + 푍푇 + 1
As a general rule, 휂푟(푇) is significantly compromised when 푢 deviated from 푠 by more than a factor of two [22].
Segmented Thermoelements 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. The compatibility factor is then a measure of maximum suitability over the segmented materials since one current should flow through the segmented thermoelement. The compatibility factor is a thermodynamic properties essential for designing an efficient segmented 5-38
thermoelectric device. If the compatibility factors differ by a factor of 2 or more, the maximum efficiency can in fact decrease by segmentation [22].
Table 5.2 Spread sheet Calculation of p-type Element Performance based on Snyder and Ursell (2003) [22, 24].
-3 T (K) Material 훼 × 10 k ZT 푢 (1/V) 푠 (1/V) 휂푟 (%) 휂 (%) (V/K) (cm) (W/mK) 973 CeFe4Sb3 156 0.849 2.689 1.04 3.62 2.82 15.66 0.00
948 CeFe4Sb3 160 0.842 2.698 1.07 3.64 2.89 16.24 0.44
923 CeFe4Sb3 164 0.834 2.706 1.10 3.66 2.97 16.83 0.90
898 CeFe4Sb3 166 0.826 2.712 1.10 3.66 3.02 17.09 1.38
873 CeFe4Sb3 167 0.818 2.717 1.10 3.64 3.07 17.19 1.89
848 CeFe4Sb3 167 0.809 2.72 1.07 3.62 3.11 17.14 2.40
823 CeFe4Sb3 167 0.8 2.722 1.05 3.59 3.15 17.07 2.93
798 CeFe4Sb3 166 0.791 2.723 1.02 3.56 3.18 16.82 3.46
773 CeFe4Sb3 164 0.782 2.723 0.98 3.51 3.20 16.40 4.01
748 CeFe4Sb3 162 0.772 2.722 0.93 3.47 3.22 15.99 4.55
723 CeFe4Sb3 160 0.762 2.721 0.89 3.43 3.25 15.55 5.09
698 CeFe4Sb3 157 0.752 2.719 0.84 3.39 3.26 14.95 5.64
673 CeFe4Sb3 154 0.741 2.716 0.79 3.35 3.27 14.36 6.18
673 Zn4Sb3 200 3.118 0.637 1.36 3.73 3.97 21.08 6.18
648 Zn4Sb3 195 3.064 0.637 1.26 3.66 3.99 20.13 6.95
623 Zn4Sb3 191 3.008 0.632 1.20 3.61 4.05 19.40 7.73
598 Zn4Sb3 187 2.949 0.622 1.14 3.55 4.14 18.74 8.50
573 Zn4Sb3 182 2.889 0.61 1.08 3.50 4.23 17.95 9.27
548 Zn4Sb3 178 2.825 0.6 1.02 3.45 4.33 17.22 10.04
523 Zn4Sb3 173 2.76 0.593 0.96 3.40 4.41 16.30 10.81
498 Zn4Sb3 168 2.691 0.591 0.88 3.36 4.45 15.30 11.56
482 Zn4Sb3 165 2.645 0.593 0.84 3.33 4.47 14.65 12.04
482 Bi2Te3 196 2.225 1.071 0.78 3.51 3.53 14.20 12.04
473 Bi2Te3 198 2.174 1.043 0.82 3.51 3.72 14.82 12.31
448 Bi2Te3 202 2.016 0.992 0.91 3.51 4.24 15.97 13.14
423 Bi2Te3 204 1.834 0.971 0.99 3.50 4.75 16.51 14.05
398 Bi2Te3 203 1.632 0.97 1.04 3.48 5.28 16.49 15.02
373 Bi2Te3 200 1.415 0.979 1.08 3.45 5.91 16.16 16.04
348 Bi2Te3 194 1.198 0.987 1.11 3.41 6.69 15.46 17.09 5-39
323 Bi2Te3 185 1.015 0.985 1.11 3.37 7.55 14.35 18.15
298 Bi2Te3 173 0.927 0.963 1.00 3.32 8.03 12.70 19.18
Three potential materials, Bi2Te3, Zn4Sb3, and CeFe4Sb3 as a segmented element are examined for adequacy. The three thermoelectric material properties along with temperature are listed in
Table 5.2. The reduced current density 푢, compatibility factors 푠, reduced efficiency 휂푟 and maximum reduced efficiency 휂푟,푚푎푥 are computed numerically as shown in the table using
Equations (5.87), (5.94), (5.92), and (5.95) with an initial 푢𝑖 = 3.6179 that has to be a value corresponding to the maximum efficiency. The computed results are plotted in Figure 5.14. The reduced current density 푢, which was defined as the ratio of the current density 푗 to the heat flow density 푘∇T, is a numerical solution from the heat balance equation of Equation (5.80). The reduced current density remains almost constant over the temperature. However, the compatibility factor 푠, which is a value corresponding to the maximum reduced efficiency, shows the maximum potential of the material. So the discrepancy between 푢 and 푠 causes the decrease of the reduced efficiency, which is shown in Figure 5.14 (b). This should be a good measure of selecting segmented materials.
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(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. These plots are based on the work of Snyder (2006) [22, 24].
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Thermoelectric Potential From Equation (5.75), the thermoelectric potential Φ is defined as
푞 1 (5.96) Φ = = 훼푇 − 푗 푢
Then,
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
q̇ = Ej = j∇Φ (5.99)
From Equation (5.88), the infinitesimal efficiency 휂 is developed as
푑Φ (5.100) 푞̇푑푥 푗∇Φdx 푑푥 ∆Φ Φ − Φ Φ 휂 = = = 푑푥 = = ℎ 푐 = 1 − 푐 푞 푗Φ Φ Φ Φℎ Φℎ
Hence, the infinitesimal efficiency is 1 (5.101) Φ 훼푐푇푐 − 푢 휂 = 1 − 푐 = 1 − 푐 Φℎ 1 훼ℎ푇ℎ − 푢ℎ 5-42
Using Equation (5.93) and (5.96), we have
푑푙푛Φ (5.102) 휂 = 푇 푟 푑푡
Taking integration of Equation (5.102) gives
Φ 푇ℎ 휂 (5.103) ℎ = 푒푥푝 [− ∫ 푟 푑푇] Φ 푇 푐 푇푐
From Equation (5.101), the overall efficiency of a finite segment is given by
푇ℎ 휂 (5.104) 휂 = 1 − 푒푥푝 [− ∫ 푟 푑푇] 푇 푇푐
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. The resulting maximum efficiency is given by
푇 − 푇 √1 + 푍푇 − 1 (5.105) 휂 = ℎ 푐 푚푎푥 푇 푇 ℎ √1 + 푍푇 + 푐 푇ℎ
This equation is the normally attained maximum conversion efficiency in Chapter 2 with assumption of the constant material properties.
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Problems
5.1. Derive Equation (5.26) and (5.28) in detail. 5.2. Describe the Thomson heat. 5.3. What is the Thomson relations? 5.4. Derive Equations (5.65) and (5.66) of the exact solutions and explain what is the meaning of the exact solution. 5.5. Derive Equations (5.69) and (5.70) of the ideal equation and explain what is the meaning of the ideal equation. 5.6. Derive in detail Equation (5.94). 5.7. 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. 5.9. (half term project) Develop a Mathcad program to provide Figure 5.12 for a thermoelectric cooler and explore the uncertainty of the ideal equation comparing with the exact solution.
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References
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20. Shi, X., et al., Multiple-filled skutterudites: high thermoelectric figure of merit through separately optimizing electrical and thermal transports. J Am Chem Soc, 2011. 133(20): p. 7837-46. 21. Poudel, B., et al., High-thermoelectric performance of nanostructured bismuth antimony telluride bulk alloys. Science, 2008. 320(5876): p. 634-8. 22. Snyder, G. and T. Ursell, Thermoelectric Efficiency and Compatibility. Physical Review Letters, 2003. 91(14). 23. Baranowski, L.L., G.J. Snyder, and E.S. Toberer, Concentrated solar thermoelectric generators. Energy & Environmental Science, 2012. 5(10): p. 9055. 24. Rowe, D.M., Thermoelectrics handbook; macro to nano. 2006, Roca Raton: CRC Taylor & Francis,.