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1.2.3 Equation of Thermoelectric Efficiency
ОглавлениеThe equation for the thermoelectric efficiency of thermoelectric generators is derived as below. The electrical current density through a thermoelectric material includes the drift current density (J d) driven by the potential gradient and the thermal diffusion current density (J t) driven by the temperature gradient,
where V is the potential, T is the temperature, and σ and S are the electrical conductivity and the Seebeck coefficient of the thermoelectric material, respectively.
When a thermoelectric leg is disconnected with the external circuit, the voltage on the thermoelectric leg is the open‐circuit voltage (V OC) (Figure 1.20a). At this open‐circuit condition, the net current at any position is zero, J = 0. The drift current density is opposite to the thermal diffusion current density, and they cancel each other at any position. In terms of the Eq. (1.38), the open‐circuit voltage is given by
(1.39)
When an external load is connected to the thermoelectric leg, the charge carriers can flow through the external circuit. The current density increases with the decreasing external load resistance. When the external load resistance is zero, the voltage is zero. This is the short‐circuit condition. The current density equals the thermal diffusion current density J t (Figure 1.20b).
Figure 1.20 Steady temperature and voltage profiles of a thermoelectric leg under (a) open‐circuit and (b) short‐circuit conditions.
It should be noted that ΔT is lowered at the short circuit condition in comparison with the open‐circuit condition because of the Peltier effect. According to the Peltier effect, the current flow cools down the hot side and heats up the cold side. This will lower the J t.
In the normal case, there are both drift current density and thermal diffusion current density. In terms of the conservation law, the total amount of charge carriers in a material must be conserved. Thus, the variation of the charge carrier density (n) over time is related to the gradient of the current density,
(1.40)
By replacing J with the Eq. (1.38), it is given,
(1.41)
This indicates that the variation of charge carrier density over time can be achieved by varying the voltage or the temperature gradient over position.
The energy flux in a thermoelectric material includes the heat transport and thermoelectric conversion. J Q is used for the heat flux that is the amount of energy crossing a cross‐section area per unit time,
The first term in the right side of Eq. (1.42) is the heat conduction from the hot to cold part within a material. The second term is the heat related to the thermoelectric effects. Derivation of the Eq. (1.42) with position leads to
It is important to understand the terms in the right side of the Eq. (1.43). The second term where S varies with position is related to the Peltier and Thomson effects. As shown in Figure 1.21a, the system has a uniform temperature, and there is current flow through the system. This can be the case, right after applying a current to a system which was initially in equilibrium. Because the Seebeck coefficient (S M) of the metal electrode is different from that (S) of the thermoelectric material, the transport of the current from the left metal electrode through the contact into the thermoelectric material leads to the energy absorption (cooling) by (S M‐S)TJ due to the Peltier effect. On the other hand, the same amount of energy is released at the right contact, when the current flows from the thermoelectric material into the right metal electrode. The energy release is also owing to the Peltier effect.
Figure 1.21b shows the case when the temperature is not uniform. In addition to the Peltier effect at the two contacts, STJ is not uniform inside the thermoelectric material because S varies with temperature. The Seebeck coefficient (S a) at the position a is different from that (S b) at the position b. As a result, heat transfer takes place between the two positions because of the Thomson effect. The heat transfer (Q T) due to the Thomson effect is given by
(1.44)
Figure 1.21 Scheme of the different heat fluxes of an n‐type thermoelectric material connected to metallic contacts under the flow of an electrical current density J. (a) Only Peltier effect takes place at the junctions and there is no temperature difference. (b) Under a temperature gradient, S varies inside the thermoelectric material, which produces heat generation in the bulk (Thomson effect), the thermoelectric effects at the positions a and b are indicated. Peltier effect also takes place at the junctions.
The third term SJ ∂T/∂x in the right side of Eq. (1.43) can be converted to the potential gradient in terms of the Eq. (1.38),
where is the electric field. The first term in the right side of Eq. (1.45) is the electrical work, and the second term is the Joule heat generated in the material by the electrical current.
The last term at the right side of Eq. (1.43) is the energy variation due to the variation of the charge carrier concentration. At steady‐state condition the current becomes uniform, and this term is thus zero.
These analyses indicate that the energy flux variation in a material can include the variations of heat conduction, the Peltier effect, the Thomson effect, the Joule heating, the electrical work, and the distribution of charge carriers,
Heat flux can change the temperature of a material. Consider in the volume of Adx with A as the cross‐section area,
(1.47)
where d is the mass density and C p is the specific heat.
When a material is initially at thermal equilibrium with homogeneous temperature, its temperature can be varied only when an electrical work or heat transfer is applied to it. If an external electric field is applied to a system, the electrical work will affect the temperature of the system,
By replacing the first term in the right side of Eq. (1.48) with Eq. (1.46), the following equation is obtained,
Assume that the thermal conductivity is independent of position at steady state, (1.49) becomes
as and at steady state. The Eq. (1.50) is used for the evaluation of thermoelectric efficiency.
The input heat flux or the temperature difference between the hot and cold sides is usually constant when a thermoelectric system is evaluated. Consider that the material properties are independent of the temperature and they are identical for the p‐type and n‐type legs, the power delivered to the external load is given by
(1.51)
where I is the current through the external load and ΔV ex = IR ex is the voltage drop on the external load.
The values of I, ΔV ex, and P depend on the resistance of the external load. The resistance of the external load is usually varied from the open‐circuit voltage condition (R ex → ∞) to the short‐circuit voltage (R ex → 0) to find the optimal power on the external load.
As shown in Figure 1.22, for a system with a p‐type leg and an n‐type leg, the open‐circuit voltage is
(1.52)
when the p‐ and n‐type thermoelectric materials have the same absolute Seebeck coefficient value. When the system is connected to an external load and form a close circuit, the open‐circuit voltage dissipated on the external load and the internal resistance (R in) that includes the resistances of materials, contacts, and wires,
Figure 1.22 (a) A thermoelectric generator with n‐ and p‐type legs. Heat transfer (Qin) into the thermoelectric legs from the hot side and heat transfer (Qout) out from the cold side. (b) Temperature and voltage profiles at open‐circuit condition.
(1.53)
The current through the circuit can be obtained as
Thus, the equation for the power is given by
By introducing a parameter the equation (1.55) becomes
When R in = R ex, the external load has the maximum power (P max),
(1.57)
Figure 1.23 shows the variations of the output power with the external load resistance at different temperature differences [34]. The output power increases when the temperature difference increases. However, at the optimal output power, the corresponding load resistance is independent of the temperature difference.
The equation for the efficiency (η) can be obtained as it is the ratio of the output power to the incident heat (Q in) into the system,
(1.58)
Heat energy transport through this system with the two thermoelectric legs, p‐ and n‐type legs. The incident heat at the hot side is balanced by the Peltier effect at the contacts between the legs and the metal electrodes, and the heat transfers due to the temperature gradient. The heat transferring from the hot to the cold side is equivalent to the Peltier effect at the contacts and the heat out from the cold side. Thus, the equations at the two boundaries (hot and cold side) are given by
Figure 1.23 Variations of the power output of a TEG with the load resistance at ΔT = 10 and 20 K.
Source: Madan et al. [34]. © American Chemical Society.
(1.60)
By ignoring the Thomson effect, the equation (1.50) for one leg becomes
(1.61)
Using the boundary conditions of T(0) = T H and T(L) = T C, the following equations can be obtained,
(1.62)
(1.63)
At x = 0,
(1.64)
Using Eq. (1.54) in Eq. (1.59), the expression for Q in can be obtained as below,
where is the intrinsic resistance of one leg.
In terms of Eqs. (1.54), (1.56), and (1.65), the conversion efficiency is given by
where z is the figure of merit of the thermoelectric module,
(1.67)
The internal resistance (R in) includes the contact resistances and the resistances of the thermoelectric legs. When the contact resistances are neglected, z becomes the figure of merit (Z) of the materials,
(1.68)
In terms of the Eq. (1.66), the m value corresponding to the maximum efficiency is
(1.69)
where is the mean operation temperature,
(1.70)
The maximum efficiency (η max) is given by
(1.71)
This equation is the same as the Eq. (1.32). This efficiency is always less than the Carnot efficiency (η c),
(1.72)
