1. Governing equations
A Peltier module is characterized by three lumped parameters: its Seebeck
coefficient \(S\) (volts per kelvin), internal electrical
resistance \(R_m\), and thermal conductance \(K_m\)
between its two faces. Given a drive current \(I\), hot-side
temperature \(T_h\), and cold-side temperature \(T_c\)
(all in kelvin), the heat pumped from the cold side and rejected at the hot
side are:
$$ Q_c = S I T_c - \tfrac{1}{2} I^2 R_m - K_m (T_h - T_c) $$
$$ Q_h = S I T_h + \tfrac{1}{2} I^2 R_m - K_m (T_h - T_c) $$
Cold-side heat absorbed and hot-side heat rejected. The middle term is
resistive (Joule) heating, split evenly between both faces; the last term is passive
back-conduction through the module.
The electrical power drawn by the module is the difference between these two,
which must equal \(V \cdot I\):
$$ P_{in} = Q_h - Q_c = S I (T_h - T_c) + I^2 R_m $$
Energy conservation check: heat rejected always equals heat absorbed plus
electrical power in.
Coefficient of performance for cooling is simply the useful heat moved per
watt spent moving it:
$$ \mathrm{COP}_{cooling} = \frac{Q_c}{P_{in}} $$
2. Extracting the module parameters
TEC1-12706 datasheets don't publish \(S\), \(R_m\),
or \(K_m\) directly — they publish four rated numbers instead:
maximum voltage, maximum current, maximum heat pumped, and maximum achievable
\(\Delta T\), all measured at a reference hot-side temperature
\(T_{h,ref}\). Typical published figures for this part:
| Rated value |
Symbol |
Typical figure |
| Max voltage |
Vmax |
15.4 V |
| Max current |
Imax |
6.4 A |
| Max heat pumped (ΔT = 0) |
Qmax |
92 W |
| Max ΔT (Qc = 0) |
ΔTmax |
66 K |
| Reference hot-side temp |
Th,ref |
298 K (25°C) |
These are commonly published figures for the TEC1-12706 — actual values vary
by manufacturer and batch, so this should be treated as a representative
model, not a spec for this exact unit.
These four ratings correspond to two known operating points: \(Q_c=0\)
at \(\Delta T = \Delta T_{max}\), and \(\Delta T = 0\)
at \(Q_c = Q_{max}\), both driven at \(I_{max}\).
Solving the governing equations at both points simultaneously gives closed-form
expressions for the three module parameters:
$$ S = \frac{Q_{max} + \tfrac{1}{2} I_{max} V_{max}}{I_{max}\left(T_{h,ref} + \tfrac{1}{2}\Delta
T_{max}\right)} $$
$$ R_m = \frac{V_{max} - S\,\Delta T_{max}}{I_{max}} $$
$$ K_m = \frac{S I_{max} T_{c,ref} - \tfrac{1}{2} I_{max}^2 R_m}{\Delta T_{max}} $$
Tc,ref = Th,ref − ΔTmax = 232 K.
Plugging in the datasheet numbers:
| Parameter |
Value |
| Seebeck coefficient, S |
0.0667 V/K |
| Internal resistance, Rm |
1.72 Ω |
| Thermal conductance, Km |
0.967 W/K |
Sanity check
Feeding these three values back into the governing equations at both original
reference points reproduces Qmax = 92 W and ΔTmax = 66 K
to within rounding — confirming the three parameters are self-consistent with
the datasheet ratings before using them anywhere else.
3. Steady-state performance
The prototype doesn't yet have calibrated thermocouples on the hot and cold
faces (that's Rev D), so the numbers here use a representative operating
point rather than measured data: \(I = 5.85\,\mathrm{A}\) per
module (consistent with ~23.4A total across 4 parallel modules, close to the
measured 23.8A), cold face at \(T_c = 283\,\mathrm{K}\) (10°C)
and hot face at \(T_h = 303\,\mathrm{K}\) (30°C) — a 20 K spread
across the module, a reasonable middle-of-the-road assumption for a loaded
TEC with an actively cooled hot side.
$$ Q_c = (0.0667)(5.85)(283) - \tfrac{1}{2}(5.85)^2(1.72) - (0.967)(20) \approx 61.7\ \mathrm{W} $$
$$ P_{in} = (0.0667)(5.85)(20) + (5.85)^2(1.72) \approx 66.7\ \mathrm{W} $$
$$ \mathrm{COP}_{TEC} = \frac{61.7}{66.7} \approx 0.93 $$
| Quantity |
Per module |
× 4 modules |
| Heat absorbed, Qc |
61.7 W |
≈ 247 W |
| Electrical power, Pin |
66.7 W |
≈ 267 W |
| COPTEC |
≈ 0.93 |
267W modeled at the module level leaves about 18W unaccounted for against
the measured 285W at the wall — a plausible split between the 0.7A fan
(≈8.4W), the small water pump (≈4–5W typical for this class), and buck
converter / wiring losses. Dividing the cooling power by the full wall-plug
power instead of just the TEC power gives a more honest system-level number:
\(\mathrm{COP}_{system} = 247/285 \approx 0.87\). That's the
number that actually matters for the "low-power" goal.
4. Transient response of the cold plate
The electrical time constant of a TEC module is on the order of milliseconds
— negligible next to the thermal mass of the aluminum heatsink it's bolted
to. So the cold plate's temperature over time can be modeled as a lumped
thermal capacitance receiving constant cooling power \(Q_c\)
while losing heat back to the incoming airstream by forced convection:
$$ m c_p \frac{dT}{dt} = -Q_c + hA\,(T_\infty - T) $$
m = heatsink mass, cp = specific heat of aluminum (897 J/kg·K), hA
= convective conductance to the airstream, T∞ = ambient intake temperature.
This is a first-order linear ODE with a standard exponential solution:
$$ T(t) = T_{ss} + \left(T_0 - T_{ss}\right) e^{-t/\tau}, \qquad T_{ss} = T_\infty - \frac{Q_c}{hA},
\qquad \tau = \frac{m c_p}{hA} $$
Backing \(hA\) out of the measured 5–8°C ambient-to-outlet drop
(using the low end, 5.5 K, against the modeled
\(Q_{c,total} \approx 247\,\mathrm{W}\))
gives \(hA \approx 44.9\,\mathrm{W/K}\). Assuming the shared cold
heatsink is roughly 0.6 kg of aluminum:
$$ \tau = \frac{(0.6)(897)}{44.9} \approx 12.0\ \mathrm{s} $$
The cold plate should reach ~63% of its final temperature drop in about 12
seconds, and ~95% within roughly three time constants (~36 seconds) of power-on.
Testable prediction
This is a falsifiable number — clocking time-to-steady-state with a
thermocouple and stopwatch on the actual unit is a direct check on whether
the assumed heatsink mass and convective coefficient are close to reality.
Flagged for Rev D.
5. What the numbers say
A modeled COP of roughly 0.9 means the array is moving somewhat less heat
than it consumes in electrical power — not unreasonable for a thermoelectric
device fighting a 20 K spread, but it's roughly 3–4x below what a small
vapor-compression loop typically achieves (COP of roughly 2–4, even with a
compressor in the 30–60W class). At 285W continuous draw, this also directly
undercuts the project's founding premise: a device meant to help during grid
outages shouldn't need a small generator to run.
There's also a design detail worth flagging independent of the TEC-vs-
compressor question: the cold-side fan pushes air along the
heatsink fins — an axial-flow path — rather than through them, the
more common cross-flow layout. Letting the boundary layer grow uninterrupted
along the full fin length instead of being repeatedly disrupted likely lowers
the effective convective coefficient compared to a conventional through-flow
heatsink — a plausible reason the measured 5–8°C outlet drop is smaller than
the ~20 K the model predicts at the cold face itself.
Whether a small vapor-compression system closes the efficiency gap enough to
justify its added complexity — refrigerant handling, compressor noise and
vibration, reduced portability — against a COP roughly 3–4x higher and a
fraction of the power draw, is the open question for Rev D onward.