The math against stacking Peltiers
In practical set ups, Peltiers start to lose effectiveness when the temperature difference between hot and cold sides exceeds around 20K. The seemingly obvious solution if higher temperature differences are desired is to “stack” Peltiers, but the internet warns against this for good reason.
While the conclusion is right, the reasons are often misunderstood. This article shows the simple math behind stacking Peltiers and the effect of the accumulated waste heat, which does in general make them unsuitable for stacking. But understanding the math also allows us to identify special cases where is could work, such as the butter cooling application.
Knowing the math also means ideas can be explored first by modelling prior to building up a system, and confirming expected results which helps to identify issues with real systems, such as a bad thermal joint.
First, lets return to the basic Peltier circuit and review the available temperature and “short circuit” power of a single Peltier. If these concepts and the electrical analogy for thermal circuits is unclear, make sure to review this article again.
As discussed previously, the heart of the Peltier is a core temperature difference (Tc) that is proportional to the current. But the waste heat (W) flowing through the thermal resistance on the hot side of the Peltier effectively reduces the available temperature (Ta) according to the simple formula:
Ta = Tc - WX (K or °C)
Both the core temperature and waste heat are functions of the current (Tc = KI; W = I²R) , but for now it makes sense to assume a constant current, which then fixes Tc and W.
The maximum available power can then be calculated using this reduced available temperature (Ta) divided by the total thermal resistance of the Peltier. In the previous article these were identified as Xc and Xh, in this article we will just use “X” for each side, giving a total of 2X for the whole Peltier (as can be seen in the circuit):
Pa = Ta / 2X = (Tc - WX) / 2X
For a TEC 12708 running at 3A, using typical values values as previously discussed (K = 20, R = 1.5Ω, X = 0.9K/W), we could expect Tc = 60K, Ta = 48K and Pa = 26.6W (cooling), with an input of 13.5W, and a COP of around 2.0.
Now let’s do the same math but with two stacked Peltiers, which appear in series in the thermal circuit.
Our instinct is to say that we should simply double up: double the core temperature, double the waste heat, and double the total thermal resistance from hot to cold. In this case we expect the available temperature to double, while the available power (for cooling) remains the same:
Ta = (2Tc - 2WX) = 2 (Tc - WX), (double)
Pa = (2Tc - 2WX) / 4X = (Tc - WX) / 2X (same as before).
This is similar to say putting two batteries in series, the voltage doubles, the internal resistance also doubles, and the short circuit current remains the same. Thus for two stacked TEC 12708 both run at 3A, we expect Ta = 96K (which would be great!) the same maximum cooling of Pa = 26.6W, but two lots of waste heat (2 x 13.5W), the COP drops to about 1.0.
But you would be wrong. The catch: if you look closely, the waste heat of the Peltier #1 has to travel through three lots of half thermal resistances “X” as shown by the purple line. The waste heat of Peltier #2 is the same, travelling through only one half. Thus the reduction in the available temperature is 3 + 1 = four times as much, not double as might have been expected. This means:
Ta = 2Tc - 4WX
It’s an important point to realise here that this is not about Peltier #2 having to “work harder” to “get rid” of the waste heat of Pelter #1, but rather simply reducing the available temperature due to the longer path for the waste heat of Peltier #1 to travel to get outside. For stacked Peltiers, there is nothing that can be done about this: larger heatsinks or heat exchanges on the hot side will do nothing to eliminate this effect, it’s fundamental to the Peltier. It’s also nothing to do with the Peltiers being close together or “leaking” heat or a myriad of other explanations. It’s just the result of the waste heat of Pelter #1 travelling through Peltier #2 and creating a temperature drop, just like a current flowing through a longer electrical wire creates a larger voltage drop.
For the available power, the total circuit resistance is now double, meaning we have:
Pa = (2Tc - 4WX) / 4X
Which means the available cooling power will be less than before, and keep in mind we now have twice the input power. The actual values for two stacked TEC 12708 both run at 3A, are Ta = 71.4K (which is still better than a single Peltier) while Pa drops to 19.8W in the input power doubles to 27W, resulting in a COP of just 0.7. Actually in many applications 0.7 would be fine, but given that this is under ideal conditions (infinite heatsinks on both sides), practical set ups would likely be much lower.
If we expand the analysis to three or more Peltiers, it should be possible to see that the effect gets even worse. It turns out that for “n” Peltiers, the available temperature and power is:
Ta = (nTc - n²WX)
Pa = (nTc - n²WX)/2nX = (Tc - nWX)/2X
The “n²” factor in the Ta formula really takes off with 3 or more Peltiers, and the numbers start to get gruesome. For example, a four stack has a Ta less than a single Pelter (45.6K) and a maximum cooling power of just 6.3W with 54W input (again based on a TEC 12708, 3A input, K = 20, I = 1.5). And the actual numbers are worse due to the thermocouple effect, which is excluded here for simplicity.
The butter cooler (warmer): and example where stacked Peltiers worked much better than a single Peltier
This all sounds bad, but actually there are cases where stacking can be OK. The above analysis covers the two extreme cases: the “open circuit” available temperature (Ta), and the “short circuit” power (Pa) with both hot and cold sides at the same temperature with zero thermal resistance to ambient (infinite heatsinks). And it’s also running at fairly high levels (3A for an 8A TEC), where waste heat is significant.
The butter cooler/warmer application (pictured, to be covered by a future article), the stacked Peltier actually works around 5 times better than a single Peltier. The reason is threefold: low power (just 0.75W), only requires about 10K temperature difference and there’s no forced cooling so the thermal resistances outside the Peltier are fairly high. As will be shown in that article, the math just works out.
Once the math is understood, it’s also possible to start playing with some other configurations. For example, it’s not necessary to run both Peltiers at the same current, and there can be advantages to running Peltier #1 at a lower current than Peltier #2, to reduce the impact of the waste heat from Peltier #1, while still getting a higher temperature range than a single Peltier.
Another option is to have a single cold side Peltier thermally linked to two Peltiers in parallel on the hot side. In this case the waste heat of Peltier #1 splits before travelling through to the outside. And you could again play around with the currents of the three Peltiers in the system to find better performance. It’s planned to explore this option in a future article.
Finally, it’s noted that some manufacturers have pre-assembled stacked Peltiers. These have yet to be tested, but they should benefit from having reduced thermal resistance, with one less layer of ceramic and one less sloppy thermal joint made by us, getting thermal paste everywhere.
Conclusion
An example of using Circuitlab to model a fridge application, using the 1+2 configuration
In general stacking Peltiers does not work in high power applications, but can be suitable for low power applications. The Peltier thermal circuit used in this website gives engineers and DYI enthusiasts the ability to calculate if an idea might work or benefit from stacking, prior to going through the effort of assembling an actual system.
Simply lay out the components in a circuit, and either calculate or use a circuit simulator, and play with the current or configurations.