Flexible Thermoelectric Polymers and Systems. Группа авторов. Читать онлайн. Hotlib. HOTLIB.NET

the thermoelectric legs. When the contact resistances are neglected, z becomes the figure of merit (Z) of the materials,

(1.68) upper Z equals StartFraction upper S squared sigma Over kappa EndFraction period

In terms of the Eq. (1.66), the m value corresponding to the maximum efficiency is

(1.69) m Subscript eta Baseline equals StartRoot 1 plus z upper T overbar EndRoot

where upper T overbar is the mean operation temperature,

(1.70) upper T overbar equals one half left-parenthesis upper T Subscript normal upper H Baseline plus upper T Subscript normal upper C Baseline right-parenthesis period

The maximum efficiency (η _max) is given by

(1.71) eta Subscript max Baseline equals StartFraction upper Delta upper T Over upper T Subscript normal upper H Baseline EndFraction StartStartFraction StartRoot 1 plus z upper T overbar EndRoot minus 1 OverOver StartRoot 1 plus z upper T overbar EndRoot plus StartFraction upper T Subscript normal upper C Baseline Over upper T Subscript normal upper H Baseline EndFraction EndEndFraction period

This equation is the same as the Eq. (1.32). This efficiency is always less than the Carnot efficiency (η _c),

(1.72) eta Subscript normal c Baseline equals StartFraction upper Delta upper T Over upper T Subscript normal upper H Baseline EndFraction period

1.3 Peltier Cooling

The Peltier effect can be used for thermoelectric cooling. The Peltier cooling principle is quite different from the conventional cooling technologies that usually use compressed liquid or gas. Because of the compressor and circuit fluid, the conventional cooling instruments like air conditioners and refrigerators are quite bulky, heavy, and noisy. No moving part is required for a Peltier cooler; it is thus quiet and can have a very small size.

As shown in Figure 1.24, when an electrical current passes through the thermoelectric legs, both the electrons in the n‐type leg and holes in the p‐type leg transport from the cold side to the hot side. The charge carriers bring heat from the cold side to the hot side and thus cool down the cold side. The heat pumped at the cold side is Π×I, where Π is the Peltier coefficient and I the current density. Π is related to the Seebeck coefficient by the Kelvin relationship, Π = S×T.

Apart from the Peltier cooling, Joule heat is generated by the electrical current, and heat is transferred from the hot side to the cold side due to the temperature gradient. Half of the overall Joule heat transports to the cold side or the hot side. By neglecting the Thomson effect, the heat absorption at the cold side of a thermoelectric leg is given by

(1.73) upper Q equals upper S upper T Subscript upper C Baseline upper I minus one half upper I squared upper R minus upper K upper Delta upper T comma

Schematic illustration of thermoelectric cooling. An external electricity source is connected to the two legs, and electrical current flows from the n-type leg to the p-type leg. The curved arrows indicate the transportation of the charge carriers from the cold side to the hot side in the thermoelectric legs.

Figure 1.24 Schematic illustration of thermoelectric cooling. An external electricity source is connected to the two legs, and electrical current flows from the n‐type leg to the p‐type leg. The curved arrows indicate the transportation of the charge carriers from the cold side to the hot side in the thermoelectric legs.

where R is the electrical resistance of the thermoelectric leg, K the thermal conductance of the thermoelectric material, and ΔT = T _H − T _C.

The input power by the electricity includes the heat pumping and the Joule heat,

(1.74) upper P Subscript i n Baseline equals upper S upper Delta upper T upper I plus upper I squared upper R period

The ratio of the heat absorbed at the cold side to the input electrical power is the energy efficiency of cooling. It is called as the coefficient of performance (COP),

(1.75) upper C upper O upper P equals StartFraction h e a t a b s o r b e d Over e l e c t r i c a l p o w e r i n p u t EndFraction equals StartFraction upper S upper T Subscript upper C Baseline upper I minus one half upper I squared upper R minus upper K Subscript t h Baseline upper Delta upper T Over upper S upper Delta upper T upper I plus upper I squared upper R EndFraction period

As shown in Eq. (1.73), the rate of heat pumping is a quadratic equation of the current I. There is a maximum rate of heat pumping. The corresponding current at the maximum heat pumping rate is given by

(1.76) upper I Subscript max Baseline equals StartFraction upper S upper T Subscript c Baseline Over upper R EndFraction period

At this maximum cooling rate, the corresponding COP is given by

(1.77) upper C upper O upper P Subscript max Baseline equals StartFraction one half upper Z upper T Subscript upper C Superscript 2 Baseline minus upper Delta upper T Over upper Z upper T Subscript normal upper H Baseline upper T Subscript normal upper C Baseline EndFraction period

The electrical current at the maximum cooling rate is different from that corresponds to the maximum COP. The electrical current corresponding to the maximum COP is given by

(1.78) upper I prime Subscript max Baseline equals StartFraction upper S upper Delta upper T Over upper R StartRoot left-parenthesis 1 plus upper Z upper T overbar EndRoot minus 1 EndFraction period

The maximum COP is shown by the Eq. (1.79),

(1.79)

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