PromLegion DX4085, Руководство пользователя

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PromLegion Ltd Z-Meter DX4085/ User Guide
9. APPENDIX 1. THEORETICAL GROUNDINGS
9.1. Time Constant
Let us consider a single-stage TE module. The ambient temperature
is
𝑇𝑇
𝑚𝑚
. At a certain moment electric current is applied to the module.
The differential equation describing transient dynamics for a pellet of
the TE module can be presented as the following exponential
superposition:
∆𝑇𝑇(𝑡𝑡, 𝑥𝑥) = � 𝐴𝐴
𝑎𝑎
𝑈𝑈
𝑎𝑎
(𝑥𝑥)𝑒𝑒
−𝑚𝑚
𝑛𝑛
𝑎𝑎
+ ∆𝑇𝑇
𝑠𝑠𝑎𝑎
(𝑥𝑥)
∞
𝑎𝑎=1
(5.1.1)
where
∆𝑇𝑇(𝑡𝑡, 𝑥𝑥) = 𝑇𝑇 − 𝑇𝑇
𝑚𝑚
,
𝑇𝑇
- temperature of the pellet point located at a time
𝑡𝑡
and a generalized coordinate
𝑥𝑥
,
𝑈𝑈
𝑎𝑎
,
𝑚𝑚
𝑎𝑎
- the eigen functions and eigen-values,
𝐴𝐴
𝑎𝑎
- thermal amplitudes,
∆𝑇𝑇
𝑠𝑠𝑎𝑎
(𝑥𝑥)
- stationary
∆𝑇𝑇
value.
The solution (5.1.1) analysis yields that the cooling process can be
divided into two stages: irregular and regular. The first one is dictated
by the initial moment's conditions and is described by a multi-
exponential interference. This phase fades out rather quickly and in
case TEC pellets thermal conductance is high enough, the temporal
behavior can be characterized by the only exponent, i.e for all
possible n:
𝑚𝑚
𝑚𝑚𝑎𝑎𝑎𝑎
≪ 𝑚𝑚
𝑎𝑎
(5.1.2)
The theory yields the following expression for the time constant
𝜏𝜏 = 1 𝑚𝑚
𝑚𝑚𝑎𝑎𝑎𝑎
⁄
of a single-stage TE modules:
page 50 of 64 ver.1.00 (2017)