Thermoelectric cooler efficiency and heatsink thermal impedance

Thermoelectric coolers (TECs) are common and (look) simple, but simple (and usefully accurate) design models and equations for them are less common. This design model has served well in a variety of applications, and its input needs only numbers provided in typical TEC datasheets. Though with a simplification of TEC physics, it’s realistic and accurate enough to be useful. It predicts TEC thermal load temperature (T) as a function of TEC data sheet parameters, drive current (I), thermal load power dissipation, thermal conductivity, heatsink thermal impedance, and ambient temperature (T3).

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The model is summarized in a single second-order equation for T = TEC output temperature.

T = (-P I + I2 Rp/2 + Q1)/(C1 + Cp) + Zh(Q1 + I2 Rp) + T3

Where:

P (Watts/Amp) = Peltier constant = (Qmax + Imax2 Rp/2)/Imax
Qmax (Watts) = maximum heat transfer across zero delta T (from TEC datasheet)
Imax = current for max cooling with perfect (Zh = 0) heatsink (from TEC datasheet)
Vmax = TEC voltage drop at Imax (from TEC datasheet)
Rp = TEC resistance = Vmax/Imax
Q1 = heat produced by thermal load
C1 (W/°C) = thermal conductivity of thermal load to ambient
Cp = TEC thermal conductivity = Qmax/DeltaTmax
DeltaTmax = max cooling with Imax and perfect heatsink (from TEC datasheet)
Zh (oC/W) = heatsink thermal impedance to ambient
T3 = ambient temperature

For a typical example of how this math applies to a real TEC, consider the Laird Thermal Systems 430007-509:

Qmax: 3 W

Imax: 1.5 A

Vmax: 3.4 V

Delta Tmax: 67°C

Then:

Rp = 3.4/1.5 = 2.27

P = 3 + 1.5 * 3.4 / 2 = 5.55 / 1.5 = 3.7 W/A

Cp = 3 W/67°C = 0.0448 W/°C

A useful relationship quantified by the design model math is the effect of heatsink thermal impedance on the optimum TEC drive current that generates maximum cooling. It results when the T equation is differentiated with respect to I and then solved for the maximum at dT/dI = 0. It yields:

Io = (P Zh-1)/{Rp[Zh-1 + 2(C1 + Cp)]}

Io(Zh-1) is plotted for the Laird TEC in Figure 1 (black) with the corresponding maximum Delta T (blue). Note how both curves trend to zero as Zh-1 is reduced. This effect is mainly due to the fact that the I2Rp heat dissipated by the TEC must be dumped to ambient by the heatsink, which raises its temperature, and therefore that of the TEC, in direct proportion to Zh.

Figure 1 TEC max-cooling drive current (black) and resulting cooling (blue) as functions of heatsink thermal admittance (Zh-1).

Even in situations when TEC cooling ability remains adequate and DeltaT constant, the effect on TEC current draw and power consumption is dramatic, as illustrated in Figure 2 for an example DeltaT of 40oC (Q1 and C1 = 0).

Figure 2 TEC current draw I (black) versus heatsink thermal admittance (Zh-1) for constant 40°C DeltaT.

Note that current consumption increases by 63% and power by 165% as Zh-1 declines from 1.0 to 0.13 W/°C.

Stephen Woodward’s relationship with EDN’s DI column goes back quite a long way. Over 100 submissions have been accepted since his first contribution back in 1974.

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Thermoelectric cooler efficiency and heatsink thermal impedance

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