SaM - Coupling with the Environment (and Solution)

We have seen how to model a thermal system, so let’s take an example with a target $(x)$, our sensors with which we measure the temperature $T_S$, all coupled with the ambience temperature $\left(T_A\right)$:
We want to measure $T_X$, but we can only access it via $T_S$. Suppose at regime $T_S$ reaches the value $T_S{}_{\infty }$, the complete formula is:$T_S(t)=T_S{}_{\infty }\left(1-e^{-\frac{t}{\tau }}\right)+T_S\left(0^{-}\right)e^{-\frac{t}{\tau }}$Here’s the graph:

If we perform the calculations, such that:$\frac{T_X-T_S}{R_{XS}}+\frac{T_A-T_S}{R_{SA}}-C_S{T}_S\cdot s+C_S{T}_S(0^{-})=0$We will find:$\begin{array}{l}{T}_S{}_{\infty }=\frac{T_A{R}_{XS}}{R_{XS}+R_{SA}}+\frac{T_S{R}_{SA}}{R_{XS}+R_{SA}}\\ \tau =C_S\frac{R_{XS}{R}_{SA}}{R_{XS}+R_{SA}}\end{array}$==So if we have $R_{SA}\gg R_{XS}$, then we will find: $T_S{}_{\infty }\approx T_A$==. (*Often we wait $5\tau$, so in this case for $R_{SA}\gg R_{XS}$ we would need to wait at least $5C_S{R}_{XS}$)

IMPORTANTE We have defined the thermal reisistance in two ways, depending if the thermal conduction happens via motion of matter (heat convection) or not (heat conduction). Here’s a table for the value for the thermal conductivity ($K$):

	Still Air	Glass	Gold	Diamond
$K\left(\left[\frac{\text{W}}{\text{°K}\cdot \text{m}}\right]\right)$	$10^{-2}$	$1$	$10^2$	$10^3$

• Still Air means: “A porous material filled with Air”, so we are talking about only convection, no conduction.
• A bigger coefficient means a lower thermal resistance, so a better coupling, and a faster heat transfer.
  So “still air” is an thermal insulator

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Calculations: